32 The entropy window of Hastings’ area-law proof
Hastings’ proof of the one-dimensional area law [ 240 ] argues by contradiction from a cut of large entropy. This chapter records the setup of that argument, which uses no Hamiltonian: for an arbitrary unit vector on the chain of \(L\) sites of local dimension \(q\), the entropy of a cut changes by at most \(\log q\) per site, so a cut of entropy above \(S_{\max }\) is followed by a window of cuts of entropy at least \(2S_{\max }/3\); on that window the largest entropy \(S_l\) of an interval of \(l\) sites obeys \(0\le S_l\le l\log q\) and \(S_{2l}\le 2S_l\). Sites are numbered \(0,\ldots ,L-1\), the cut at \(k\) is the interval \(\lbrack 0,k)\), and \(S(A)\) is the entropy of the reduced state of a region \(A\).
Let \(\Omega \) be a unit vector on the chain of \(L\) sites. For \(i\le k\),
The cut \(\lbrack 0,k)\) is the disjoint union of \(\lbrack 0,i)\) and \(\lbrack i,k)\). The Araki–Lieb inequality gives \(S(\lbrack 0,i))\le S(\lbrack 0,k))+S(\lbrack i,k))\), and the interval \(\lbrack i,k)\) has at most \(k-i\) sites, so \(S(\lbrack i,k))\le (k-i)\log q\). This is the first step of the proof of the main theorem in [ 240 ] .
Let \(S_{\max }\ge 0\) and let \(l_0\) be a natural number with \(l_0\log q\le S_{\max }/3\), for instance \(l_0=\lfloor S_{\max }/(3\log q)\rfloor \) when \(q\ge 2\). If \(S(\lbrack 0,i)){\gt}S_{\max }\), then every \(k\) with \(i\le k\le i+l_0\) satisfies
By Theorem 32.1, \(S(\lbrack 0,k))\ge S(\lbrack 0,i))-(k-i)\log q{\gt}S_{\max }-l_0\log q\ge 2S_{\max }/3\). The source takes \(l_0=S_{\max }/(3\log q)\); the window length is a number of sites, so here it is any natural number at most this value. No condition at the right end of the chain is needed: a cut \(\lbrack 0,k)\) with \(k\ge L\) is the whole chain and has entropy zero, so for \(S_{\max }{\gt}0\) the conclusion itself forces \(i+l_0{\lt}L\).
For the window \(\lbrack i,i+l_0)\) and a length \(l\), the window entropy \(S_l\) is the largest entropy \(S(\lbrack a,a+l))\) of an interval of \(l\) sites inside the window, that is over \(a\) with \(i\le a\) and \(a+l\le i+l_0\). When no such interval exists, that is when \(l{\gt}l_0\), we set \(S_l=0\). When \(l\le l_0\) the maximum is attained by some interval inside the window. The window of [ 240 ] consists of the sites numbered \(i-1,\ldots ,i+l_0-1\) here. The window \(\lbrack i,i+l_0)\) leaves out its leftmost site and consists of the \(l_0\) sites between the cuts at \(i\) and \(i+l_0\); every cut from \(i\) to \(i+l_0\) carries the bound of Theorem 32.2. The argument uses only intervals of at most \(l_0\) sites whose cuts carry this bound, so the omitted site plays no role.
For a unit vector and every length \(l\),
The first two bounds hold for the entropy of every region of at most \(l\) sites. For the third, when \(2l\le l_0\), take an interval \(\lbrack a,a+2l)\) inside the window of largest entropy; it is the disjoint union of the intervals \(\lbrack a,a+l)\) and \(\lbrack a+l,a+2l)\), both inside the window, and subadditivity gives \(S_{2l}\le S(\lbrack a,a+l))+S(\lbrack a+l,a+2l))\le 2S_l\). When \(2l{\gt}l_0\) the left side is zero. The source defines \(S_l\) only for \(l\le l_0\); setting \(S_l=0\) for \(l{\gt}l_0\) is a convention adopted here, under which \(S_{2l}\le 2S_l\) holds for every \(l\).
Let \(\xi '\), \(C_1\) and \(C_2\) be real numbers. Suppose that for every interval \(\lbrack a,a+2l)\) inside the window,
Then for every \(l\) with \(2l\le l_0\),
This is the step from eq. (relent) to eq. (claim) of [ 240 ] , lines 201–204 and 234–242; the bound on each pair of adjacent intervals is a hypothesis.
Take an interval \(\lbrack a,a+2l)\) inside the window of largest entropy. The two intervals \(\lbrack a,a+l)\) and \(\lbrack a+l,a+2l)\) lie inside the window, so each has entropy at most \(S_l\), and the hypothesis for this pair gives the bound.
32.1 The doubling argument in Hastings’ area law
The proof of the area law in [ 240 ] bounds the length of a window on which every interval adjacent to a fixed cut carries large entropy. Let \(S_l\) be the largest entropy of an interval of \(l\) sites in a window of length \(l_0\). The argument uses only \(0\le S_l\le l\ln D\) and the strengthened doubling inequality, and iterates it from a starting length. The printed iteration loses two factors of two, recorded in [ 207 ] ; the statements below carry the corrected constants.
For real \(\xi '\), \(C_1\), \(C_2\) and \(D\) put
where \(\lceil u\rceil _+=\max \{ 0,\lceil u\rceil \} \) is the least natural number at least \(u\). If \(\xi '{\gt}0\) and \(2C_1{\gt}1\), then \(\xi _0{\gt}0\).
Since \(2C_1{\gt}1\), \(\ln (2C_1){\gt}0\), so \(\xi _0=2\xi '\ln (2C_1){\gt}0\).
Let \(\xi '{\gt}0\), \(2C_1{\gt}1\) and \(l_1\ge \xi _0\). For every natural \(N\),
and if moreover \(\ln C_1+C_2\ge 0\), then
Put \(x=2C_1\) and \(r=e^{-l_1/\xi '}\). Since \(l_1/\xi '\ge 2\ln x\), we have \(rx^2\le 1\), so \(r{\lt}1\). From \(n+1\le 2^n\) we get \(e^{-2^nl_1/\xi '}\le r^{n+1}\), and the geometric series gives the sum bound \(r/(1-r)\). The first inequality reduces to \(r(1+x\ln x)\le 1\), which follows from \(rx^2\le 1\) and \(x\ln x\le x(x-1)\le x^2-1\). The second inequality is the geometric sum \(\sum _{n}2^{-n}\le 2\) together with \(l_1\ge \xi _0\).
Let \(S\) be a real sequence indexed by the natural numbers, let \(l_0\), \(\xi '{\gt}0\), \(C_1\), \(C_2\) and \(D\) be real with \(2C_1{\gt}1\) and \(\ln C_1+C_2\ge 0\). Assume \(S_l\le l\ln D\) whenever \(l\le l_0\), and
whenever \(2l\le l_0\). Let \(l_1\) be a natural number with \(l_1\ge \xi _0\) and let \(l=2^Nl_1\le l_0\). Then
This is the bound of [ 240 , lines 221–224 ] with the decrease per doubling corrected, as recorded in [ 207 ] .
Dividing (1) by \(2l\) shows that one doubling lowers \(S_l/l\) by \(1/(2\xi ')\), up to the errors \(C_1e^{-l/\xi '}/\xi '\) and \((\ln C_1+C_2)/(2l)\). By induction on \(N\), starting from \(S_{l_1}\le l_1\ln D\), the quotient \(S_l/l\) is at most \(\ln D-N/(2\xi ')\) plus the two error sums along \(2^nl_1\) for \(n{\lt}N\). Lemma 32.1.2 bounds these sums by \(1/\xi _0\) and \((\ln C_1+C_2)/\xi _0\).
Under the hypotheses of Theorem 32.1.3, assume moreover \(S_l\ge 0\) whenever \(l\le l_0\). Then
This replaces the bound \(\xi '\ln (2C_1)\, 2^{\lceil \xi '\ln D +(2+C_2)\xi '/\xi _0+1/2\rceil }\) of [ 240 , line 229 ] ; the exponent is doubled and the prefactor is the starting length, as recorded in [ 207 ] .
Suppose \(l_0\ge l_12^M\) with \(l_1=\lceil \xi _0\rceil \). Since \(\ln C_1{\lt}\ln (2C_1)\), we have \(\ln C_1/\xi _0{\lt}1/(2\xi ')\), and the choice of \(M\) makes \(\ln D-M/(2\xi ')+(2+\ln C_1+C_2)/\xi _0\) negative. Theorem 32.1.3 at \(N=M\) then gives \(S_{l_12^M}{\lt}0\), contradicting positivity of entropy.
32.2 The contradiction argument in Hastings’ area law
The main theorem of [ 240 ] bounds the entropy of every initial segment of the chain by a constant \(S_{\max }\) that does not depend on the length of the chain. Its proof runs by contradiction. If an initial segment had entropy above \(S_{\max }\), then removing sites one at a time would keep the entropy of every cut in a window of length \(l_0\), where \(S_{\max }=3l_0\ln D\), above \(2S_{\max }/3\). On that window the strengthened doubling inequality holds, and the doubling argument of Section 32.1 bounds \(l_0\) above, which contradicts the choice of \(S_{\max }\). The statements below carry out this argument with the window facts and the doubling inequality as hypotheses. The value of \(S_{\max }\) is the one forced by the corrected doubling iteration of [ 207 ] .
With \(\xi _0\) and \(M\) as in Definition 32.1.1, put
If \(D\ge 1\), then \(S_{\max }\ge 0\).
Both \(l_0\) and \(\ln D\) are nonnegative when \(D\ge 1\).
Let \(\xi '{\gt}0\), \(2C_1{\gt}1\), \(D\ge 1\) and \(C_2\ge -2\). Then
and if moreover \(C_1\ge 1\), then
The source’s bound \(S_{\max }=c_0\xi '\ln \xi '\ln D\, 2^{\xi '\ln D}\) of [ 240 , equation (smd) ] has the base \(2\) in place of \(4\). The larger base is the correction recorded in [ 207 ] . Requiring \(C_1\ge 1\) loses nothing, since enlarging \(C_1\) only weakens Lemma 1 of the source.
Since \(M\le 2(\xi '\ln D+(2+C_2)\xi '/\xi _0+1/2)+1\) and \(\lceil \xi _0\rceil _+\le \xi _0+1\), the first bound follows from the definition. For \(C_1\ge 1\) we have \(\ln (2C_1)\ge \ln 2\), so \((2+C_2)\xi '/\xi _0=(2+C_2)/(2\ln (2C_1))\le (2+C_2)/(2\ln 2)\).
For a unit vector on a chain of \(L\) sites with local dimension \(q\ge 1\), the entropy of the first \(i\) sites is at most \((L-i)\ln q\).
The complement of the first \(i\) sites is the set of the remaining \(L-i\) sites. A pure state has equal entropies on complementary regions, and the entropy of a region of \(L-i\) sites is at most \((L-i)\ln q\).
Let \(\Omega \) be a gapped ground vector of a finite-range Hamiltonian on a chain of \(L\) sites with local dimension \(q\ge 1\). Let \(\xi '{\gt}0\), \(2C_1{\gt}1\) and \(\ln C_1+C_2\ge 0\), and let \(l_0\) and \(S_{\max }\) be as in Definition 32.2.1 with \(D=q\). Write \(S_k\) for the entropy of the first \(k\) sites, and let \(W_i(l)\) be real numbers, standing for the largest entropy of an interval of \(l\) sites in the window of length \(l_0\) starting at site \(i\). Assume:
\(S_i-(k-i)\ln q\le S_k\) for every \(i\) and \(k\) with \(i\le k\le L\);
\(0\le W_i(l)\le l\ln q\) for every \(i\) and \(l\) with \(l\le l_0\) and \(i+l_0\le L\);
for every \(i\) with \(i+l_0\le L\) such that \(S_k\ge 2S_{\max }/3\) for all \(k\) with \(i\le k\le i+l_0\),
\begin{align*} W_i(2l)\le 2W_i(l)-\bigl(1-2C_1e^{-l/\xi '}\bigr) \frac{l}{\xi '}+\ln C_1+C_2 \end{align*}whenever \(2l\le l_0\).
Then \(S_i\le S_{\max }\) for every \(i\). This is a stage of the argument: the three items are hypotheses. Once they are established for the ground vector, it is the main theorem of [ 240 ] , equation (entbd), with \(S_{\max }\) corrected as in [ 207 ] .
Suppose \(S_i{\gt}S_{\max }\). If \(i+l_0{\gt}L\), then Theorem 32.2.3 gives \(S_i\le (L-i)\ln q\le l_0\ln q\le S_{\max }\), a contradiction. Otherwise the first item gives \(S_k\ge S_{\max }-l_0\ln q=2S_{\max }/3\) for every \(k\) with \(i\le k\le i+l_0\). The second and third items then are the hypotheses of Theorem 32.1.4 for the sequence \(W_i\), which gives \(l_0{\lt}\lceil \xi _0\rceil _+2^M=l_0\).
32.3 The three-way splitting in Hastings’ approximate ground-state projector
The proof of Lemma 1 of [ 240 ] begins by splitting the Hamiltonian at a cut into a left part, a buffer part and a right part, and by centering each part in the ground vector. This section carries out that step for a finite-range Hamiltonian of range \(R\) on the chain of \(L\) sites. The source works with nearest-neighbour terms and a buffer at distance \(l/3\) from the cut; here the buffer width is an arbitrary natural number \(m\). Norms are operator norms.
Let \(H=\sum _Xh_X\) be a finite-range Hamiltonian of range \(R\) on the chain of \(L\) sites, let \(j\) be a cut, so that the sites \(i{\lt}j\) lie on its left, and let \(m\) be a natural number. The left part \(H_L\) is the sum of the terms \(h_X\) with \(X\subseteq \{ 0,\dots ,j-m-1\} \), the right part \(H_R\) is the sum of the terms with \(X\subseteq \{ j+m,\dots ,L-1\} \), and the buffer part \(H_B\) is the sum of all other terms. Each part is Hermitian. For a vector \(\Psi _0\) and a matrix \(M\), the centering of \(M\) in \(\Psi _0\) is \(\widetilde M=M-\operatorname {Re}\langle \Psi _0,M\Psi _0\rangle \, 1\); it is additive in \(M\), and Hermitian when \(M\) is. For a Hermitian \(M\) the expectation \(\langle \Psi _0,M\Psi _0\rangle \) is real. These are the operators of [ 240 , lines 774–782 ] , with \(l/3\) replaced by \(m\).
For every cut \(j\) and every \(m\), \(H=H_L+H_B+H_R\). Let \(\Psi _0\) be a unit vector with \(H\Psi _0=E_0\Psi _0\). Then each centered part has expectation zero in \(\Psi _0\), and
If moreover \(H-E_0\ge \Delta (1-|\Psi _0\rangle \langle \Psi _0|)\), then \(\Psi _0\) is a gapped ground vector of \(\widetilde H_L+\widetilde H_B+\widetilde H_R\) with energy zero and the same gap \(\Delta \). With the convention of [ 240 , lines 766–767 ] that the ground energy is zero, this is the identity \(H=H_L+H_B+H_R\) for the centered parts of [ 240 , lines 779–782 ] .
Every support is left, right or buffer, and no support is both left and right, since a support is nonempty and \(j-m\le j+m\). Summing the terms over the three classes gives \(H=H_L+H_B+H_R\). Centering is additive, so the centered parts sum to \(H-\operatorname {Re}\langle \Psi _0,H\Psi _0\rangle \), and \(\langle \Psi _0,H\Psi _0\rangle =E_0\). The expectation of \(\widetilde M\) is \(\langle \Psi _0,M\Psi _0\rangle - \operatorname {Re}\langle \Psi _0,M\Psi _0\rangle =0\) for a Hermitian \(M\). The gap condition is unchanged, since the shifted operator differs from \(H\) by the scalar \(E_0\).
The left part acts as the identity off \(\{ 0,\dots ,j-m-1\} \), the right part off \(\{ j+m,\dots ,L-1\} \), and the buffer part off the window \(\{ j-m-R,\dots ,j+m+R-1\} \). Centering preserves each of these supports. In particular the left and right parts lie at distance at least \(m\) from the cut, for every \(m\), as the parts \(H_L\) and \(H_R\) of [ 240 , lines 774–778 ] lie at distance \(l/3\).
A sum of terms supported in a set is supported in that set, and the identity is supported everywhere. A buffer support \(X\) contains a site \(x\ge j-m\) and a site \(y{\lt}j+m\); since \(X\) has diameter at most \(R\), every site \(z\in X\) satisfies \(j-m-R\le z{\lt}j+m+R\).
A sum of \(k\) terms has norm at most \(kJ\). At most \(2^{2R+1}\) supports of range \(R\) contain a given site, and there are at most \(2(m+R)\, 2^{2R+1}\) buffer supports. Hence, for \(J\ge 0\),
linearly in the buffer width. For a unit vector \(\Psi _0\), centering at most doubles the norm, so \(\| \widetilde H_B\| \le 4(m+R)\, 2^{2R+1}J\).
The first bound is the triangle inequality. A support containing a site \(x\) lies in the \(2R+1\) sites within distance \(R\) of \(x\), which have at most \(2^{2R+1}\) subsets. Every buffer support contains a site of the window \(\{ j-m-R,\dots ,j+m+R-1\} \), which has at most \(2(m+R)\) sites. Finally \(|\operatorname {Re}\langle \Psi _0,M\Psi _0\rangle |\le \| M\| \) for a unit vector \(\Psi _0\), and the identity has norm at most one.
32.4 Hastings’ approximate factorization of the ground-state projector
Lemma 1 of [ 240 ] writes the projector \(P_0\) onto the ground state of a gapped finite-range chain approximately as a product \(O_BO_LO_R\) of three operators of norm at most one: \(O_L\) acts on the sites left of a cut, \(O_R\) on the sites right of it, and \(O_B\) on the \(2l\) sites around it. This section records the steps of the proof in the order of the appendix of [ 240 ] . The first step, the splitting of the Hamiltonian at the cut into a left, a buffer and a right part, is Section 32.3. Throughout, \(H\) is Hermitian with ground energy \(E_0\), every other eigenvalue is at least \(E_0+\Delta E\) with \(\Delta E{\gt}0\), and norms are operator norms. As in the source, the sites are numbered \(1,\dots ,N\), \(X_{a,b}\) is the set of sites \(i\) with \(a\le i\le b\), and the cut \(j\) separates \(X_{1,j}\) from \(X_{j+1,N}\); on the chain of \(L\) sites numbered \(0,\dots ,L-1\) of Section 32.3, where the cut \(j\) has the sites \(i{\lt}j\) on its left, this is the shift of every site down by one, with \(N=L\), so \(X_{1,j}\) becomes \(\{ 0,\dots ,j-1\} \) and \(X_{j+1,N}\) becomes \(\{ j,\dots ,L-1\} \). The main theorem of [ 240 , lines 124–144 ] uses the decay length
where \(v\) and \(\xi _C\) are the velocity and length scale of the Lieb–Robinson bound [ 240 , lines 96–112 ] . The source puts \(\xi '=6\xi \). For a fixed length \(\xi \), the argument below gives the factorization at every rate \(\xi '{\gt}6\xi \) but does not reach \(\xi '=6\xi \) itself, so in this chapter \(\xi '\) denotes any number with \(\xi '{\gt}6\xi \), except in Lemma 32.4.16, which is stated as printed with \(\xi '=6\xi \), and in Theorem 32.4.14, which reaches \(\xi '=6\xi \) for a length obtained by raising the velocity.
From the filtering on, \(H_L\), \(H_B\), \(H_R\) denote the parts of Section 32.3 centered by their ground-state expectations, \(H_\bullet -\langle \Psi _0,H_\bullet \Psi _0\rangle \), as the source does before filtering them [ 240 , lines 781–782 ] . Centering subtracts a multiple of the identity, which commutes with \(H\) and is fixed by the filter below. It therefore changes neither \([H,H_\bullet ]\) nor \(\widetilde H_\bullet -H_\bullet \), and the bounds on the localized parts below hold for the centered and the uncentered parts alike.
Gaussian-filtered operators
For \(q{\gt}0\) and a matrix \(W\), the Gaussian-filtered operator is
the Heisenberg evolution of \(W\) averaged against the centered Gaussian of variance \(q/\Delta E^2\). For \(q=0\) the Gaussian is the point mass at \(t=0\) and \(\widetilde W=W\). These are the operators \(\tilde H_L^0\), \(\tilde H_B^0\), \(\tilde H_R^0\) of [ 240 , lines 783–793 ] for \(W\) one of the three centered parts of the Hamiltonian.
Let \(v\) and \(w\) be eigenvectors of \(H\) with eigenvalues \(E\) and \(E'\). Then
so \(\widetilde W\) and \(W\) have the same matrix elements between ground vectors. Filtering is additive in \(W\), does not increase the norm, and returns \(H\) when \(W=H\); in particular, if \(H=\sum _iH_i\) then \(\sum _i\widetilde H_i=H\), as in [ 240 , line 793 ] . For a ground vector \(\psi \), the component of \(\widetilde W\psi \) orthogonal to the ground space has norm at most \(\| W\| e^{-q/2}\| \psi \| \).
In the eigenbasis of \(H\) the evolution multiplies the matrix element between energies \(E\) and \(E'\) by \(e^{i(E-E')t}\), and the Gaussian average of this phase is \(e^{-q((E-E')/\Delta E)^2/2}\). Each conjugate of \(W\) has the norm of \(W\), and \(H\) commutes with its own evolution. An excited eigenvector has energy at least \(\Delta E\) above a ground vector, so its coefficient in \(\widetilde W\psi \) carries a factor at most \(e^{-q/2}\).
Let \(\psi \) be a ground vector of \(H\) such that \(W\psi \) is orthogonal to the ground space. Then
In particular this holds when \(\Psi _0\) is a unique gapped ground vector and \(\langle \Psi _0,W\Psi _0\rangle =0\), as for each of the centered parts \(H_L\), \(H_B\), \(H_R\). For these parts this is [ 240 , lines 794–815 ] : the bound depends on \(W\) only through its commutator with \(H\), which lives near the boundary of its support.
Expand in the eigenbasis of \(H\). The coefficient of \([H,W]\psi \) on an eigenvector of energy \(E\) is \((E-E_0)\) times that of \(W\psi \). Ground coefficients of \(W\psi \) vanish, and for an excited eigenvector \(E-E_0\ge \Delta E\) while the filter contributes at most \(e^{-q/2}\). Summing the squared coefficients gives the bound.
Let \(q{\gt}0\), \(T\ge 0\) and \(\varepsilon \ge 0\), and suppose that \(\| [e^{iHt}We^{-iHt},O]\| \le \varepsilon \) for all \(|t|\le T\). Then
For \(q=0\) the filtered operator is \(W\), and the hypothesis at \(t=0\) gives \(\| [W,O]\| \le \varepsilon \). With the Lieb–Robinson bound for \(\varepsilon \) and \(T\) proportional to \(l\), this is the locality of the filtered parts used at [ 240 , lines 816–828 ] .
Split the time integral at \(|t|=T\). On \([-T,T]\) the commutator is at most \(\varepsilon \) and the Gaussian has mass at most one. Outside, the commutator is at most \(2\| W\| \, \| O\| \), and the Gaussian of variance \(h=q/\Delta E^2\) gives the complement of \([-T,T]\) mass at most \(e^{-T^2/(2h)}\), by the shift \(t=T+s\) and \((T+s)^2\ge T^2+s^2\).
For a set \(K\) of sites, let \(\mathbb E_K\) be the normalized partial trace over the sites outside \(K\), tensored with the identity there. Let \(q{\gt}0\), \(T\ge 0\) and \(\varepsilon \ge 0\). If \(\| e^{iHt}We^{-iHt}-\mathbb E_K(e^{iHt}We^{-iHt})\| \le \varepsilon \) for all \(|t|\le T\), then
The operator \(\mathbb E_K(\widetilde W)\) acts on \(K\) and has norm at most \(\| W\| \). For \(q=0\) the filtered operator is \(W\), and the hypothesis at \(t=0\) gives the bound \(\varepsilon \). It replaces the truncated evolution by which [ 240 , lines 816–828 ] approximates the filtered parts.
The map \(M\mapsto M-\mathbb E_K(M)\) is linear and continuous, so it commutes with the time integral. Split the integral at \(|t|=T\) as in Theorem 32.4.4; outside \([-T,T]\) the integrand is at most \(2\| W\| \), since the partial trace does not increase the norm.
The localized parts and the cut-offs
Let \(H\) be a finite-range Hamiltonian of range \(R\) on the chain of \(L\) sites with terms of norm at most \(J\), and let \(H_L\), \(H_B\), \(H_R\) be its parts at the cut \(j\) with buffer width \(m\). Then
with \(c_R=2N_R(R+1)2^{2R+1}\), where \(N_R\) bounds the number of terms that cross a cut. The commutator \([H,H_L]\) acts on the sites \(\{ j-m-2R,\dots ,j-m+R-1\} \) and \([H,H_R]\) on \(\{ j+m-R,\dots ,j+m+2R-1\} \), intersected with the chain. The source states \(\| [H,H_L]\| \le J^2\) [ 240 , line 794 ] ; this fails already for nearest-neighbour terms, and the constant \(c_R\) is the correction recorded in [ 134 ] .
Let \(H\) be a finite-range Hamiltonian of range \(R\) on the chain of \(L\) sites with terms of norm at most \(J\), let \(H_L\), \(H_B\), \(H_R\) be its parts at the cut \(j\) with buffer width \(m\), filtered with parameter \(q{\gt}0\), and put \(h=q/\Delta E^2\). Let \(S_L=\{ j-m-2R,\dots ,j-m+R-1\} \) and \(S_R=\{ j+m-R,\dots ,j+m+2R-1\} \), intersected with the chain, be the windows on which \([H,H_L]\) and \([H,H_R]\) act, and put \(S_B=S_L\cup S_R\). For \(\ell \ge 0\) let \(K_\bullet \) be the set of sites within distance \(\ell \) of \(S_\bullet \), and
for \(\bullet \) one of \(L\), \(B\), \(R\). Then \(M_\bullet \) is Hermitian and \(M_\bullet -H_\bullet \) acts on \(K_\bullet \). With \(C'\) and \(v\) the constants of Theorem 31.2.1 for \(b_0=2^{2R+1}J\) and \(c_R\) the constant of Theorem 32.4.6, for every \(T\ge 0\)
The bounds do not depend on \(L\). This is [ 240 , lines 816–828 ] with the error measured from the filtered parts: the source prints the bound \(\mathcal O(\Delta E^{-1}J^2e^{-l/(3\xi )})\) on \(\| M_\bullet -H_\bullet \| \), which cannot hold in general, while its next step uses the distance \(\| \widetilde H_\bullet -M_\bullet \| \) [ 145 ] . Since \(\widetilde H_L+\widetilde H_B+\widetilde H_R=H-E_0\), the sum \(M=M_L+M_B+M_R\) satisfies \(\| M-(H-E_0)\| \le \| \widetilde H_L-M_L\| +\| \widetilde H_B-M_B\| +\| \widetilde H_R-M_R\| \); with the ground energy set to zero, as in [ 240 , lines 766–767 ] , this is the estimate of [ 240 , lines 849–850 ] .
Let \(\theta {\gt}0\), and let \(O_L\) and \(O_R\) be the spectral projections of the Hermitian operators \(M_L\) and \(M_R\) onto the eigenvectors whose eigenvalues have absolute value at most \(\theta \). Suppose that \(M_L\) acts on \(X_{1,j}\) and \(M_R\) on \(X_{j+1,N}\). Then \(O_L\) and \(O_R\) are commuting orthogonal projections, \(O_L\) acts on \(X_{1,j}\) and \(O_R\) on \(X_{j+1,N}\), and
With the parameters of the source, \(q=(l/3)\Delta E/v\) and the buffer width and localization radius proportional to \(l\), Theorems 32.4.3 and 32.4.7 give \(\| M_L\Psi _0\| ,\| M_R\Psi _0\| \le \mathcal O(\mathrm{poly}(l)\, e^{-l/(3\xi )})\), and the threshold \(\theta =\Delta E^{-1}J^2e^{-l/(6\xi )}\) then gives \(\| (O_L-1)\Psi _0\| ,\| (O_R-1)\Psi _0\| \le \mathcal O(\mathrm{poly}(l)\, e^{-l/(6\xi )})\), which is [ 240 , lines 829–841 ] with the polynomial factor kept. The source cuts off at the eigenvalues at most \(\theta \), not at their absolute values. The bound \(\| e^{i(M_L+M_R)t}O_LO_R-O_LO_R\| \le 2|t|\theta \) used at [ 240 , lines 879–880 ] needs the cut-off on the absolute value: with the printed cut-off, \(M_L\) may have eigenvalues far below \(-\theta \) on the range of \(O_L\) [ 184 ] .
Let \(\Psi _0\) be a gapped ground vector of a finite-range Hamiltonian \(H\) of range \(R\) with terms of norm at most \(J\ge 0\), with ground energy \(E_0\) and gap \(\Delta E{\gt}0\). Split \(H\) at the cut \(j\) with buffer width \(m\), center the left and right parts at their ground-state expectations, and localize them as in Theorem 32.4.7, with filter parameter \(q\ge 0\) and radius \(k\), where \(R+k\le m\). Let \(O_L\), \(O_R\) be their cut-offs at \(\theta {\gt}0\) and \(H_0=M_L+M_R\). Then \(M_L\) acts on \(X_{1,j}\) and \(M_R\) on \(X_{j+1,N}\), so Theorem 32.4.8 applies, and for every \(T\ge 0\)
where \(\delta \) is the bound of Theorem 32.4.7 for \(M_L\) and \(M_R\) with \(\ell =k\). Moreover \(H-E_0-H_0\) acts on the sites within \(m+2R+k\) of the cut, with
a bound that does not depend on the length of the chain. This assembles [ 240 , lines 766–842 ] .
Centering subtracts a multiple of the identity, which changes neither \([H,H_\bullet ]\) nor \(\widetilde H_\bullet -H_\bullet \). Hence \(\| M_\bullet \Psi _0\| \le \| \widetilde H_\bullet \Psi _0\| +\| \widetilde H_\bullet -M_\bullet \| \le \Delta E^{-1}e^{-q/2}c_RJ^2+\delta \) by Theorems 32.4.3 and 32.4.7, and Theorem 32.4.8 gives the first two bounds. The centered parts sum to \(H-E_0\), so \(H-E_0-H_0\) is the centered buffer part minus the two partial-trace corrections. The centered buffer part has norm at most twice \(\| H_B\| \), and each correction is at most \(\| \widetilde H_\bullet -H_\bullet \| \le \sqrt{2h/\pi }\, \| [H,H_\bullet ]\| \), by the Duhamel formula and the first absolute moment of the Gaussian.
The Gaussian time average and the boundary operator
For \(q{\gt}0\), \(P_q\) is the Gaussian time average of Definition 32.9.1. With ground energy zero, Theorems 32.9.2 and 32.9.6 give \(\| P_q(H)-P_0\| \le e^{-q/2}\) and \(\| P_q(M)-P_q(H)\| \le \sqrt q\, \Delta E^{-1}\| M-H\| \) for Hermitian \(M\), the gap bound and the first inequality of the display in [ 240 , lines 843–864 ] , with \(M=M_L+M_B+M_R\).
For \(M\) Hermitian with \(\| M-H\| \le \delta \) and commuting orthogonal projections \(O_L\), \(O_R\) with \(\| \Psi _0-O_LO_R\Psi _0\| \le \eta \), Theorem 32.10.2 gives \(\| P_q(M)\, O_LO_R-P_0\| \le e^{-q/2}+\sqrt q\, \Delta E^{-1}\delta +\eta \), which is [ 240 , lines 865–876 ] ; the cut-offs of Theorem 32.4.8 supply \(\eta \).
The interaction picture \(e^{i(A+B)t}=U(t)e^{iAt}\) and the Gaussian average \(P_B\) of \(U(t)\) are Definitions 32.11.1 and 32.11.4. With \(A=M_L+M_R\), \(B=M_B\) and \(X=O_LO_R\), Theorem 32.11.6 is the bound of [ 240 , lines 877–905 ] on \(\| P_BX-P_0\| \).
Let \(H\) be a finite-range Hamiltonian of range \(R\) on the chain of \(L\) sites with local dimension \(d\ge 1\) and terms of norm at most \(J\). Let \(H_0\), \(A\) and \(B\) be Hermitian, with \(H-H_0\) supported on a set of sites \(Z\), and let \(P_B\) be the Gaussian average of Definition 32.11.4 for \(A\), \(B\) and a variance \(h{\gt}0\). Let \(C'\) and \(v\) be the constants of Theorem 31.2.1 for \(b_0=2^{2R+1}J\). Then for every \(\ell \ge 0\) and \(T\ge 0\) there is an operator \(O_B\) supported on the sites within distance \(\ell \) of \(Z\), with \(\| O_B\| \le 1\) and
For \(T=\ell /(4v)\) the last two terms are \(\sqrt h\, C'|Z|^2\| H-H_0\| e^{-\ell /4}\) and \(4e^{-\ell ^2/(32v^2h)}\). No positivity of \(O_B\) is claimed, as in the note added to [ 240 ] .
Write \(V(t)=e^{iHt}e^{-iH_0t}\). Its generator is the Heisenberg evolution of \(H-H_0\), which Theorem 31.2.1 approximates by an operator supported within distance \(\ell \) of \(Z\) up to \(C'|Z|^2\| H-H_0\| e^{v|s|-\ell /2}\) at time \(s\). The evolution generated by the projected generator is a unitary \(\widetilde V(t)\) supported there, within \(|t|\, C'|Z|^2\| H-H_0\| e^{vT-\ell /2}\) of \(V(t)\) for \(|t|\le T\). The propagator \(U(t)\) is within \(|t|(\| A+B-H\| +\| A-H_0\| )\) of \(V(t)\). Put \(O_B=\int \widetilde V(t)\, d\gamma _h(t)\). Integrating against \(\gamma _h\) with \(\int |t|\, d\gamma _h\le \sqrt h\), and using the bound \(2\) on \(\| U(t)-\widetilde V(t)\| \) where \(|t|{\gt}T\), whose Gaussian mass is at most \(2e^{-T^2/(2h)}\), gives (2).
In the source’s application \(A=H_0=M_L+M_R\) and \(B=M_B\), so \(\| A-H_0\| =0\), while \(\| A+B-H\| \), with the ground energy set to zero, is the error of the localized parts of Theorem 32.4.7, exponentially small in \(l\) [ 240 , lines 849–850 ] . Alternatively, take \(B=H-E_0-H_0\). Adding \(E_0\) to \(A\) does not change \(P_B\), since the two phases \(e^{\pm iE_0t}\) cancel, so the theorem may be applied with \(A\) and the comparison operator both equal to \(H_0+E_0\); then \(A+B=H\) and both norms vanish. \(Z\) is the window of width proportional to \(l\) that carries \(H-H_0\). With \(h=q/\Delta E^2\), \(q\) proportional to \(l\), and \(\ell \) proportional to \(l\), the bound becomes the \(\mathcal O(\Delta E^{-1}J\sqrt q\, e^{-l/(6\xi )})\) of [ 240 , lines 896–905 ] , and \(O_B\) acts on \(X_{j-l+1,j+l}\).
The factorization and its positive version
Let \(H\) be a finite-range Hamiltonian of range \(R\) with terms of norm at most \(J\), with a unique ground state \(\Psi _0\) and a gap \(\Delta E{\gt}0\) to the first excited state, and let \(P_0=|\Psi _0\rangle \langle \Psi _0|\). Set \(\xi =8\max (2v_0/\Delta E,1)\), where \(v_0\) is the velocity in the chain localization estimate. Take the variance parameter
in the filtered parts and in the time averages. When \(\xi =2v/\Delta E\) this is \((l/3)\Delta E/v\), twice the value printed in [ 240 , line 792 ] . It gives \(q/2=l/(3\xi )\), so \(e^{-q/2}=e^{-l/(3\xi )}\), the bound used for \(\| P_q-P_0\| \) in [ 240 , line 848 ] ; the printed value gives only \(e^{-q/2}\le e^{-l/(6\xi )}\). Let \(\xi '{\gt}6\xi \). Then there is a constant \(C_1\), depending only on \(\xi '-6\xi \) and on \(R\), \(J\) and \(\Delta E\), such that for every cut \(j\) and every \(l\) there are operators \(O_B\), \(O_L\), \(O_R\) with the following properties:
\(O_L\) and \(O_R\) are commuting orthogonal projections, \(O_L\) acting on \(X_{1,j}\) and \(O_R\) on \(X_{j+1,N}\);
\(\| O_B\| \le 1\), and \(O_B\) acts on \(X_{j-l+1,j+l}\);
\(\| O_BO_LO_R-P_0\| \le C_1e^{-l/\xi '}\).
The argument of the appendix gives the error \(\mathcal O(\Delta E^{-2}J^2\sqrt q\, e^{-l/(6\xi )})\) [ 240 , lines 903–907 ] , a polynomial in \(l\) times \(e^{-l/(6\xi )}\), since the localization of the boundary average carries the factor \(\sqrt q\), proportional to \(\sqrt l\). Every rate \(\xi '{\gt}6\xi \) absorbs the polynomial, but the rate \(\xi '=6\xi \) of [ 240 , lines 124–144 ] does not follow; both the doubled \(q\) and the condition \(\xi '{\gt}6\xi \) are recorded in [ 148 ] . This is a quantitatively weaker version of Lemma 32.4.16, also without the positivity of \(O_B\); the note added to [ 240 ] (lines 912–913) records that the construction need not give a Hermitian positive \(O_B\), while its other properties hold.
Here the length is \(\xi =8\max (2v_0/\Delta E,1)\), with \(v_0\) the velocity in the chain localization estimate. The commutator bound behind that estimate is linear in the size of the support, as in the source. The quadratic factor of the localization enters with supports of at most \(6R\) sites for the parts and of \(O(R+l)\) sites for the boundary operator, where the faster decay of that term absorbs the resulting polynomial in \(l\); it changes the constant, not the length or the rate. It differs from the source’s Lieb–Robinson bound in that its prefactor grows with \(J\) and the second operator acts on one site, so \((v_0,2)\) is not shown to be a Lieb–Robinson pair in the source’s sense [ 137 ] . The constant \(C_1\) depends only on \(R\), \(J\), \(\Delta E\) and \(\xi '\): it is the same for every length of the chain, every local dimension and every gapped ground state. No polynomial bound on \(C_1\) in \(\xi \) alone is asserted here, unlike Lemma 32.4.16. Replacing \(O_B\) by \(O_B^\dagger O_B\) gives the positive version at the rate \(e^{-l/(2\xi ')}\), as in Theorem 32.16.7.
Shift \(H\) by its ground energy. For \(l\ge 12R+8\) take \(s=2l/(3\xi )\), \(\theta =e^{-s/4}\), \(T=s/\Delta E\), \(k=\lceil l/4\rceil \), \(m=R+k\) and \(\ell '=l-3R-2k\). Theorem 32.4.9 gives \(O_L\), \(O_R\) and \(H_0=M_L+M_R\). Theorem 32.10.2, with the shifted Hamiltonian in both places, bounds the time average next to \(O_LO_R\), and Theorem 32.19.1 produces \(O_B\) on the sites within \(m+2R+k+\ell '=l\) of the cut. Each of the resulting error terms is at most \((a+b\sqrt l)e^{-l/(6\xi )}\) with \(a\), \(b\) depending only on \(R\), \(J\) and \(\Delta E\); the polynomial factors are absorbed by the slack between the rates \(l/8\) and \(l/(6\xi )\le l/48\). For \(l{\lt}12R+8\) the factorization \(O_B=0\), \(O_L=O_R=1\) has error \(1\), which the constant absorbs. Theorem 32.18.3 converts the square-root bound into \(C_1e^{-l/\xi '}\) for every \(\xi '{\gt}6\xi \).
Under the assumptions of Theorem 32.4.11, let \(c_R=2(R+1)2^{2R+1}e^{2R}\), so that \(v_0=1+c_RJ\), and set
For every \(\xi '{\gt}6\hat\xi \) there is a constant \(C_1\), depending only on \(R\), \(J/\Delta E\) and \(\xi '\), such that for every cut \(j\) and every \(l\) there are operators \(O_B\), \(O_L\), \(O_R\) with the properties of Theorem 32.4.11 and \(\| O_BO_LO_R-P_0\| \le C_1e^{-l/\xi '}\). The length \(\hat\xi \), unlike the length of Theorem 32.4.11, depends on \(J\) and \(\Delta E\) only through \(J/\Delta E\), as the source’s \(\max (2v/\Delta E,\xi _C)\) with \(v\) proportional to \(J\) does [ 240 , lines 96–112, 139–142 ] . The other restrictions of Theorem 32.4.11 remain [ 137 ] .
The operators \(O_B\), \(O_L\), \(O_R\) and the error do not involve the Hamiltonian. The Hamiltonian \(H/\Delta E\) has range \(R\), terms of norm at most \(J/\Delta E\), the same ground state and gap \(1\), so Theorem 32.4.11 applies to it with the length \(8\max (2(1+c_RJ/\Delta E),1)=\hat\xi \).
Let \(\kappa {\gt}6\). There is a constant \(K\ge 0\), depending only on \(R\) and \(\kappa \), with the following property. Under the assumptions of Theorem 32.4.11, with \(\hat\xi \) the length of Theorem 32.4.12, for every cut \(j\) and every \(l\) there are operators \(O_B\), \(O_L\), \(O_R\) with the properties of Theorem 32.4.11 and
The same holds with \(O_B\) positive semidefinite, with another such constant and the rate \(e^{-l/(2\kappa \hat\xi )}\), as in Theorem 32.16.7 and [ 175 ] . So the constant of the factorization at the rate \(\xi '=\kappa \hat\xi \) is bounded by a polynomial in \(\hat\xi \), as Lemma 32.4.16 requires [ 240 , lines 258–262 ] . For the source’s nearest-neighbour terms (\(R=1\)) the constant \(K\) depends only on \(\kappa \). The rate \(\xi '=6\hat\xi \) is not reached at \(\hat\xi \) itself, and \(\hat\xi \) is not shown to come from a Lieb–Robinson pair in the source’s sense [ 137 , 148 ] .
Apply Theorem 32.4.11 to \(H/\Delta E\) at the rate \(\kappa \hat\xi \), and write \(x=J/\Delta E\). The constant is \(a+b/\sqrt{2e(1/(6\hat\xi )-1/(\kappa \hat\xi ))}\), where \(a\) is the larger of the constant term of the error budget and \(e^{(12R+8)/(6\hat\xi )}\), and \(b\) is the coefficient of \(\sqrt l\). Since \(\hat\xi \ge 16\) and \(x\le \hat\xi \), the Lieb–Robinson prefactor is at most a multiple of \(\hat\xi \) and the commutator constant at most a multiple of \(\hat\xi ^2\). So the constant term of the budget is at most a multiple of \(\hat\xi ^3\), \(e^{(12R+8)/(6\hat\xi )}\le e^{(12R+8)/6}\), and \(b\) is at most a multiple of \(\hat\xi ^2\). Finally, \(1/\sqrt{2e(1/(6\hat\xi )-1/(\kappa \hat\xi ))}=\sqrt{\hat\xi /\beta }\le \hat\xi /\sqrt\beta \) with \(\beta =2e(\kappa -6)/(6\kappa )\).
In units of the gap let \(v_0=1+c_RJ/\Delta E\) be the velocity of the chain estimate, so that \(\hat\xi =8\max (2v_0,1)\), and let \(\tilde\xi =8\max (2v',1)\) be the length for the raised velocity \(v'=\tfrac 54v_0\); thus \(\tilde\xi =\tfrac 54\hat\xi \). There is a constant \(K\ge 0\), depending only on \(R\), with the following property. Under the assumptions of Theorem 32.4.11, for every cut \(j\) and every \(l\) there are operators \(O_B\), \(O_L\), \(O_R\) with the properties of Theorem 32.4.11 and
The same holds with \(O_B\) positive semidefinite, with another such constant and the rate \(e^{-l/(12\tilde\xi )}\). This is the printed rate \(\xi '=6\xi \) of [ 240 , lines 141–142 ] for the length \(\xi =\tilde\xi \), reached by choosing the length anew and not at \(\hat\xi \). The choice is admissible because the source fixes \(\xi =\max (2v/\Delta E,\xi _C)\) only through some Lieb–Robinson pair \((v,\xi _C)\) [ 240 , lines 96–112 ] , and a bound valid for \(|t|\le l/v\) remains valid for \(|t|\le l/v'\) when \(v'\ge v\) [ 137 ] . The same holds for the chain estimate, so \(\tilde\xi \) carries the restriction of \(\hat\xi \): the chain estimate is not shown to give a Lieb–Robinson pair in the source’s sense.
Since \(6\tilde\xi =\tfrac {15}2\hat\xi \), this is Theorem 32.4.13 with \(\kappa =15/2\), together with \(\hat\xi ^3\le \tilde\xi ^3\).
Let \(B\) be a matrix and \(P\), \(Q\) orthogonal projections with \(\| BQ-P\| \le \varepsilon \). Then
No bound on \(\| B\| \) is needed. This is the bound of [ 240 , lines 916–925 ] with \(B=O_B\), \(P=P_0\) and \(Q=O_LO_R\).
Write \(BQ=P+E\) with \(\| E\| \le \varepsilon \). Since \(Q\) is self-adjoint, \(QB^\dagger BQ=(BQ)^\dagger (BQ)=(P+E^\dagger )(P+E)\), and \(P^2=P\) gives \(QB^\dagger BQ-P=E^\dagger P+PE+E^\dagger E\). The triangle inequality and \(\| P\| \le 1\) finish the proof.
The note added to [ 240 ] observes that the operator \(O_B\) constructed in the appendix need not be Hermitian or positive, and offers two ways around this. The area-law proof uses Lemma 32.4.16 only through the expectation of \(O_B\), or of the positive operator \(O_B^\dagger O_B\), in the ground state and in a product of marginals, and by Theorem 32.4.15 this expectation corollary holds at the rate \(e^{-l/\xi '}\) of Theorem 32.4.11, for every \(\xi '{\gt}6\xi \), with \(\varepsilon \) replaced by \(2\varepsilon +\varepsilon ^2\); this is the first of the two routes of the note, and it is all that the area law needs. The positive operator \(O_B^+=O_B^\dagger O_B\) of the note satisfies the factorization only with the error of Theorem 32.16.6, of order \(\sqrt{\varepsilon }\), so the positive version of Lemma 32.4.16 holds with a positive semidefinite \(O_B^+\) and the rate \(e^{-l/(2\xi ')}\) in place of \(e^{-l/\xi '}\). The note itself states both changes: it calls \(O_B^+\) positive semidefinite and bounds its error by a multiple of \(\sqrt{\varepsilon (l)}\) [ 240 , lines 932–940 ] . The entry below states the lemma as printed in the body of the paper.
This is Lemma 1 of [ 240 ] . Let \(H\) be a Hamiltonian that satisfies the finite range, finite interaction strength, and gap conditions. Then for any \(j\) and any \(l\) there exist Hermitian, positive definite operators \(O_B(j,l)\), \(O_L(j,l)\), \(O_R(j,l)\) with the following properties. First, \(\| O_B(j,l)\| \le 1\), \(\| O_L(j,l)\| \le 1\) and \(\| O_R(j,l)\| \le 1\). Second,
where \(P_0=|\Psi _0\rangle \langle \Psi _0|\) is the projector onto the ground state and \(C_1(\xi )\) is bounded by a polynomial in \(\xi \). Finally, \(O_L(j,l)\) is supported on \(X_{1,j}\), \(O_R(j,l)\) on \(X_{j+1,N}\), and \(O_B(j,l)\) on \(X_{j-l+1,j+l}\) [ 240 , lines 246–267 ] .
The argument of the appendix does not prove the lemma as printed; it gives two weaker statements. Theorem 32.4.11 gives the factorization with the bound \(\varepsilon (l)=C_1e^{-l/\xi '}\) for every \(\xi '{\gt}6\xi \), with \(C_1\) depending on \(\xi '-6\xi \), rather than at \(\xi '=6\xi \); it gives the supports and the norms as stated, and \(O_L\), \(O_R\) commuting orthogonal projections, but with an operator \(O_B\) that need not be Hermitian or positive. If \(O_B\) is replaced by \(O_B^\dagger O_B\), which is positive semidefinite, has norm at most one and keeps the support, then Theorem 32.16.6 bounds the new error by \(\mathcal O(\sqrt{\varepsilon (l)})\), which decays only at the rate \(e^{-l/(2\xi ')}\); this is the version of the note added [ 240 , lines 932–940 ] . Neither statement gives positive definite operators or a positive \(O_B\) at the rate \(e^{-l/\xi '}\). At a fixed length neither reaches the rate \(\xi '=6\xi \); that rate is reached only after the length is raised, as in Theorem 32.4.14. In units of the gap the constant is bounded by a polynomial in the length, as the lemma requires, at every rate \(\xi '=\kappa \hat\xi \) with \(\kappa {\gt}6\) (Theorem 32.4.13). The printed version does not follow from this argument.
32.5 Locality of Gaussian-filtered terms on a finite graph
The approximate local projective form of [ 239 ] replaces each term \(H_i\) of a gapped Hamiltonian by its Gaussian-filtered version \(\widetilde H_i\) (Definition 32.4.1), and then by an operator \(M_i\) of finite range. This section proves the locality of the filtered terms on a finite graph and an analogous approximation constructed by partial trace. The Hamiltonian is \(H=\sum _kh_k\), with the finite-range interaction assumptions of Theorem 28.9.11: each \(h_k\) is Hermitian and acts on a set \(X_k\) of diameter at most \(R\) and at most \(v_R\) sites, and the norms of the terms containing any one site sum to at most \(b_0\). The source’s \(M_i\) evolves \(H_i\) under the terms near \(i\) only [ 239 , lines 285–307 ] . Here \(M_i\) is the normalized partial trace of \(\widetilde H_i\) onto a neighbourhood of the support of \(H_i\), which gives a support bound, the norm bound \(\| M_i\| \le J\), and an analogous localization estimate. The radius of this neighbourhood is measured from the support of \(H_i\), rather than from \(i\). The Gaussian weight printed in eq. (mid) as \(e^{-(t\Delta E)^2/2}\) is read as \(e^{-(t\Delta E)^2/(2q)}\). Both changes are recorded in [ 153 ] .
For the locality estimates below, the parameter \(\Delta E\) may be any real number. For \(q\ge 0\) and \(\Delta E\ne 0\), define \(\widetilde W\) by averaging \(e^{iHt}We^{-iHt}\) against the centered Gaussian of variance \(q/(\Delta E)^2\). If \(q=0\) or \(\Delta E=0\), set \(\widetilde W=W\). The Gaussian tail factor in the estimates is then \(1\). For \(q{\gt}0\) and \(\Delta E{\gt}0\), this is precisely the filter of Definition 32.4.1.
Let \(l\in \mathbb N\), let \(W\) act on a set \(S\) of sites and \(B\) on one site \(y\) with \(d_G(x,y)\ge l\) for every \(x\in S\). Let \(q\ge 0\), \(\Delta E\in \mathbb R\), \(\mu \ge 0\) and \(T\ge 0\), and let \(C\) and \(v\) be the constants of Theorem 28.9.11. Then
This is eq. (lbnd) of [ 239 ] , lines 266–283, with \(T=c_1l\) and the Lieb–Robinson term in place of \(g(c_1l,l)\).
Assume that for every vertex \(x\) and integer \(r\ge 0\) at most \(K_g(r+1)^k\) vertices have graph distance exactly \(r\) from \(x\), and let \(C\), \(v\) and \(T_k\) be the constants of Theorem 28.9.19. Let \(W\) act on a set \(S\) of sites, let \(l\in \mathbb N\), and let \(N_l(S)\) be the set of sites within graph distance \(l\) of \(S\). For positive local dimension, \(q\ge 0\), \(\Delta E\in \mathbb R\) and \(T\ge 0\), there is an operator \(M\) acting on \(N_l(S)\) with \(\| M\| \le \| W\| \), Hermitian when \(W\) is, and
If \(W=H_i\) is supported within distance \(R\) of \(i\), then \(N_l(S)\) is contained in the ball of radius \(l+R\) about \(i\). Thus, for integer radii \(l_{\mathrm{proj}}\ge R\), choosing \(l=l_{\mathrm{proj}}-R\) gives the support radius used in [ 239 , lines 285–320 ] and \(\| M_i\| \le J\). The localization exponential is then \(e^{vT-(l_{\mathrm{proj}}-R)/2}\). This is an analogue of the source’s approximation estimate, with a different construction and constants; it does not identify the two quantitative estimates [ 153 ] . Every finite graph satisfies the growth assumption with \(K_g\) the number of vertices and \(k=0\); the assumption only enters the constant.
Let \(H\) be Hermitian, \(q\ge 0\) and \(\Delta E{\gt}0\). Let every eigenvalue of \(H\) equal \(E_0\) or be at least \(E_0+\Delta E\), let \(\psi \) be a ground vector, and let \(W\) and \(M\) be arbitrary operators. Then the component of \(M\psi \) orthogonal to the ground space has norm at most \((\| W\| e^{-q/2}+\| \widetilde W-M\| )\| \psi \| \). If \(\Omega \) is a unit vector spanning the ground space, then
With \(M=M_i\) as in Theorem 32.5.2 and \(q=c_1l_{\mathrm{proj}}\Delta E\), these are eq. (facld) of [ 239 , lines 322–348 ] and the bound for a unique ground state at lines 375–387.
Write \(M\psi =\widetilde W\psi +(M-\widetilde W)\psi \). The excited component of \(\widetilde W\psi \) has norm at most \(\| W\| e^{-q/2}\| \psi \| \) by Theorem 32.4.2, and that of \((M-\widetilde W)\psi \) at most \(\| \widetilde W-M\| \, \| \psi \| \). For a unique ground state the projection onto the ground space is \(\phi \mapsto \langle \Omega ,\phi \rangle \Omega \).
If \(H\) is Hermitian and \(\sum _iW_i=H\), then any operators \(M_i\) satisfy \(\| \sum _iM_i-H\| \le \sum _i\| \widetilde W_i-M_i\| \). This is the comparison of \(H\) with \(M=\sum _iM_i\) in [ 239 , lines 359–374 ] . The consequence drawn there, that the ground state of \(M\) overlaps that of \(H\) once \(N\| \widetilde H_i-M_i\| \) is small compared with \(\Delta E\), is stated only to order of magnitude.
The filtered terms sum to \(H\) by Theorem 32.4.2, so \(\sum _iM_i-H=\sum _i(M_i-\widetilde W_i)\).
Degenerate ground spaces
When the ground space is degenerate, the ground states are approximate eigenvectors of the \(M_i\) only in a basis that diagonalizes the compressions \(PM_iP\) together, where \(P\) is the orthogonal projection onto the ground space [ 239 , lines 395–491 ] . Such a basis exists approximately when these compressions nearly commute. In this subsection every eigenvalue of \(H\) equals \(E_0\) or is at least \(E_0+\Delta E\), with \(\Delta E{\gt}0\), and \(q\ge 0\). The commutator of \(\widetilde W\) with a separated operator is bounded through the partial trace of Theorem 32.5.2, so the term \(N_Mg(c_1l,l)\) of eq. (lc) is replaced by the localization term of that theorem [ 153 ] . The translation invariant case [ 239 , lines 1603–1642 ] , the example at lines 1655–1676, and the summed bound eq. (oibu), which the source states only to order of magnitude, are not formalized.
For all operators \(W\) and \(B\), \(P\widetilde WP=PWP\), \(\| (1-P)\widetilde WP\| \le \| W\| e^{-q/2}\), \(\| P\widetilde W(1-P)\| \le \| W\| e^{-q/2}\), and
With \(W=M_i\) and \(B=M_j\) this is the triangle inequality of [ 239 , lines 1562–1593 ] .
The first identity holds because \(\widetilde W\) and \(W\) have the same matrix elements between ground vectors. The second bound is the bound on the excited component of \(\widetilde W\psi \) applied to \(\psi =Px\), and the third follows by taking adjoints, since the adjoint of \(\widetilde W\) is the filtered operator of \(W^\dagger \). Writing \(Q=1-P\),
and \(\| P\| \le 1\).
Under the assumptions of Theorem 32.5.2, let \(W\) act on \(S\) and \(B\) on a set disjoint from \(N_l(S)\). Assume \(q{\gt}0\). Then
for every \(T\ge 0\). At \(q=0\), the bound holds with \(\widetilde W=W\) and the Gaussian tail factor replaced by \(1\). This is an analogue of the separated-operator bound in [ 239 , lines 1548–1561 ] . Since \(N_l(S)\) is a closed neighbourhood, the support separation must exceed \(l\); if it is at least an integer \(s\ge 1\), one may choose \(l=s-1\).
The partial trace \(\mathbb E_{N_l(S)}(\widetilde W)\) acts on \(N_l(S)\) and so commutes with \(B\). Hence \([\widetilde W,B]=[\widetilde W-\mathbb E_{N_l(S)}(\widetilde W),B]\), whose norm is at most \(2\| B\| \) times the bound of Theorem 32.5.2.
Under the assumptions of Theorem 32.5.6, with \(H=\sum _kh_k\) gapped as above,
At \(q=0\), replace the Gaussian tail factor by \(1\) and set \(\widetilde W=W\). For \(l{\gt}0\), choose \(c_1{\gt}0\) with \(vc_1{\lt}1/2\), and set \(T=c_1l\) and \(q=c_1l\Delta E\). The right-hand side of (9) becomes
For fixed prefactors this decays exponentially in \(l\). With \(W=M_i\) and \(B=M_j\), it is an analogue of the low-energy estimate in [ 239 , lines 1562–1601 ] , with the different localization constants and separation convention described above.
Let \(A\) be self-adjoint with \(Av=rv\) and \(Aw=sw\) for real \(r\ne s\). Then for every operator \(B\),
With \(A=PM_jP\), \(B=PM_iP\) and \(v\), \(w\) unit ground vectors, this is the bound \(o(i)\le \| [PM_iP,PM_jP]\| /\delta \lambda _j\) of [ 239 , lines 1678–1690 ] on the off-diagonal elements of \(M_i\) in a basis of the ground space that diagonalizes \(M_j\).
Since \(A\) is self-adjoint and \(r\) is real, \(\langle v,[A,B]w\rangle =(r-s)\langle v,Bw\rangle \), and \(|\langle v,[A,B]w\rangle |\le \| [A,B]\| \, \| v\| \, \| w\| \).
32.6 One level of Hastings’ hierarchical projection
Section II of [ 239 ] builds a state close to a ground state \(\Psi _a\) of a gapped Hamiltonian from local spectral projections. Each pair operator \(M=M_{k,k+1}\) nearly has \(\Psi _a\) as an eigenvector, \(\lVert (M-c)\Psi _a\rVert \le \epsilon \) with \(c=\langle \Psi _a,M\Psi _a\rangle \). The first level projects onto the eigenvectors of \(M\) with eigenvalue within \(x\) of \(c\); each later level projects the normalized output of the previous one with the pair operator compressed to the range of the level below, the window still centred at \(c\), and the source states a recurrence for the deficiencies over \(h=\log _2N\) levels. This section proves the estimates of one level and bounds the printed scalar recurrence. Applying that recurrence to the entire hierarchy requires a further argument. The deficiency of a level is measured in norm: the source states it as an expectation, which alone would not give the normalization bound it uses at the next level [ 138 ] . Norms of operators are operator norms.
The supersite grouping, the final energy bound, the count of the final states and the extensions to higher dimensions ( [ 239 , lines 609–628 and 709–756 ] ) are not part of this section. The bound over all \(h\) levels is in Section 32.7.
For a Hermitian matrix \(M\) and real numbers \(c\) and \(x\), the window projection \(P_x(M,c)\) is the spectral projection of \(M\) onto its eigenvalues \(\lambda \) with \(\lvert \lambda -c\rvert \le x\), that is the spectral cut-off of \(M-c\) at \(x\). This is the projection \(P^1_k\) of [ 239 , lines 640–642 ] for \(c=\langle \Psi _a,M\Psi _a\rangle \).
For a Hermitian matrix \(M\) and a real number \(c\), the matrix \(M-c\) is Hermitian, and every window projection of \(M\) is an orthogonal projection.
The scalar \(c\) is real, and the spectral cut-off of a Hermitian matrix is an orthogonal projection.
Let \(P\) be an orthogonal projection and \(\Psi \) a unit vector. Then \(\langle \Psi ,P\Psi \rangle =\lVert P\Psi \rVert ^2=1-\lVert \Psi -P\Psi \rVert ^2\). If \(\lVert \Psi -P\Psi \rVert \le \delta \), then \(\langle \Psi ,P\Psi \rangle \ge 1-\delta ^2\) and \(\langle \Psi ,P\Psi \rangle \ge 1-\delta \). For every matrix \(P'\) and every vector \(\Psi \),
The vectors \(P\Psi \) and \(\Psi -P\Psi \) are orthogonal. With \(d=\lVert \Psi -P\Psi \rVert \), the expectation \(1-d^2\) is nonnegative, so \(d\le 1\) and \(d^2\le d\le \delta \). For the last bound, \(\Psi -PP'\Psi =(\Psi -P\Psi )+P(\Psi -P'\Psi )\) is a sum of orthogonal vectors, and \(P\) does not increase norms.
Let \(M\) be Hermitian, \(c\) real, \(x{\gt}0\) and \(P=P_x(M,c)\). For every vector \(\Psi \),
and for a unit vector \(\Psi \), \(\langle \Psi ,P\Psi \rangle \ge 1-\lVert (M-c)\Psi \rVert /x\) and \(\langle \Psi ,P\Psi \rangle \ge 1-(\lVert (M-c)\Psi \rVert /x)^2\). With \(\lVert (M-c)\Psi _a\rVert \le \epsilon \) and \(x=\epsilon ^{1/h}\) this is the bound \(\langle \Psi _a,P^1_k\Psi _a\rangle \ge 1-b_1\), \(b_1=\epsilon ^{1-1/h}\), of [ 239 , lines 645–657 ] , with the deficiency \(\lVert \Psi _a-P^1_k\Psi _a\rVert \le b_1\) in norm.
Let \(M\) be Hermitian, \(c\) real and \(x{\gt}0\). If a nonzero vector \(\Psi \) satisfies \(\lVert (M-c)\Psi \rVert {\lt}x\lVert \Psi \rVert \), then \(M\) has an eigenvalue \(\lambda \) with \(\lvert \lambda -c\rvert \le x\). This is the conclusion of [ 239 , lines 658–667 ] that at least one state survives the projection.
Otherwise \(P_x(M,c)=0\), and (13) gives \(\lVert \Psi \rVert \le \lVert (M-c)\Psi \rVert /x{\lt}\lVert \Psi \rVert \).
Let \(\Psi \) be a unit vector. For every vector \(v\), \(\lVert v/\lVert v\rVert -\Psi \rVert \le 2\lVert v-\Psi \rVert \). For an orthogonal projection \(Q\), \(\lVert Q\Psi /\lVert Q\Psi \rVert -\Psi \rVert ^2\le 2\lVert \Psi -Q\Psi \rVert ^2\). If \(P\) and \(P'\) are commuting orthogonal projections with \(\lVert \Psi -P\Psi \rVert \le b\) and \(\lVert \Psi -P'\Psi \rVert \le b\), then
This is \(\lVert \Psi ^1_a-\Psi _a\rVert \le 2b_1\) of [ 239 , lines 686–689 ] . A vector \(0/0\) is read as \(0\).
The first bound follows from \(\lvert 1-\lVert v\rVert \rvert \le \lVert v-\Psi \rVert \). For the second, with \(0{\lt}t=\lVert Q\Psi \rVert \le 1\), the left side is \(2-2t\) and the right side is \(2-2t^2\). If \(Q\Psi =0\), the two sides are \(1\) and \(2\). For the third, \(PP'\) is an orthogonal projection, and (12) gives \(\lVert \Psi -PP'\Psi \rVert ^2\le 2b^2\).
Let \(P\) and \(Q\) be orthogonal projections with \(PQ=P\), let \(\Psi \) be a unit vector and \(\Psi '=Q\Psi /\lVert Q\Psi \rVert \). Then
The projection of level \(m\) lies below the product of the two projections of level \(m-1\) [ 239 , lines 671–676 ] , and this bound combines the deficiencies of consecutive levels.
With \(t=\lVert Q\Psi \rVert \) and \(s=\lVert P\Psi '\rVert \), both at most one, \(\lVert P\Psi \rVert =ts\), and the claim is \(1-t^2s^2\le (1-t^2)+(1-s^2)\), that is \((1-t^2)(1-s^2)\ge 0\). If \(Q\Psi =0\), then \(P\Psi =0\) and both sides are at least one.
For every matrix \(N\) and vectors \(\phi \), \(\psi \), \(\lVert N\phi \rVert \le \lVert N\psi \rVert +\lVert N\rVert \, \lVert \phi -\psi \rVert \). For every matrix \(M\) and real \(c\), \(\lVert M-c\rVert \le \lVert M\rVert +\lvert c\rvert \), which is at most \(2\lVert M\rVert \) when \(\lvert c\rvert \le \lVert M\rVert \); for a unit vector \(\psi \), \(\lvert \operatorname {Re}\langle \psi ,M\psi \rangle \rvert \le \lVert M\rVert \). The source bounds the residual of \(\Psi ^1_a\) by \(\epsilon +2b_1J\, l_{\mathrm{proj}}\) [ 239 , lines 690–695 ] , which is the first bound with \(\lVert M-c\rVert \le J\, l_{\mathrm{proj}}\). In general the factor is \(\lVert M-c\rVert \le 2\lVert M\rVert \).
Write \(N\phi =N\psi +N(\phi -\psi )\), and use \(\lVert 1\rVert \le 1\) and the Cauchy–Schwarz inequality.
Let \(M\) be Hermitian, \(c\) real and \(x{\gt}0\). If \(\lVert (M-c)\psi \rVert \le \epsilon \) and \(\lVert \phi -\psi \rVert \le \eta \), then
With \(\psi =\Psi _a\), \(\phi =\Psi ^1_a\) and \(\eta =2b_1\) this is the estimate behind the deficiency \(b_2\) of [ 239 , lines 690–699 ] . The source’s second-level projection restricts the spectral window of \(QMQ\) to the range of \(Q=P^1_kP^1_{k+2}\) (lines 673–685). Its corresponding estimate is Theorem 32.6.13.
Let \(Q\) be an orthogonal projection, \(M\) a matrix, \(c\) real and \(\phi \) a vector with \(Q\phi =\phi \). Then \(\lVert (QMQ-c)\phi \rVert \le \lVert (M-c)\phi \rVert \). At the second level the source projects with \(P^1_kP^1_{k+2}M_{k+1,k+2}P^1_{k+2}P^1_k\), and \(\Psi ^1_a\) lies in the range of \(P^1_kP^1_{k+2}\) [ 239 , lines 673–689 ] .
Since \(Q\phi =\phi \), \((QMQ-c)\phi =Q(M-c)\phi \), and \(Q\) does not increase norms.
Let \(Q\) be an orthogonal projection, \(M\) Hermitian and \(c,x\) real. Set \(P=QP_x(QMQ,c)\). This is the spectral window of the restriction of \(QMQ\) to \(\operatorname {ran}Q\), extended by zero on \(\ker Q\). It is the projection used at later levels in [ 239 , lines 673–685 ] .
With the hypotheses of Definition 32.6.11, \(W=P_x(QMQ,c)\) commutes with \(Q\). The matrix \(P=QW\) is an orthogonal projection with \(PQ=P\). If \(Q\phi =\phi \), then \(P\phi =W\phi \); if \(Q\phi =0\), then \(P\phi =0\). These statements hold even when \(|c|\le x\), in which case the ambient projection \(W\) includes \(\ker Q\).
The identity \(Q^2=Q\) gives \((QMQ)Q=Q(QMQ)\). Functional calculus preserves this commutation, so \(WQ=QW\). The product of commuting orthogonal projections is an orthogonal projection, and \(PQ=QWQ=Q^2W=P\). Also \(P\phi =WQ\phi \), giving both action identities.
Let \(Q\) be an orthogonal projection, \(M\) Hermitian, \(c\) real and \(x{\gt}0\). Put \(P=QP_x(QMQ,c)\). For every \(\phi \) with \(Q\phi =\phi \),
If also \(\lVert (M-c)\psi \rVert \le \epsilon \) and \(\lVert \phi -\psi \rVert \le \eta \), then
No membership condition is imposed on \(\psi \). Taking \(\psi =\Psi _a\), \(\phi =\Psi ^1_a\) and \(\eta =2b_1\) gives the second-level estimate with the window centred at \(c=\langle \Psi _a,M\Psi _a\rangle \) as in [ 239 , lines 673–699 ] .
Since \(Q\phi =\phi \), \(P\phi =P_x(QMQ,c)\phi \). Apply the one-level estimate to \(QMQ\) and use \(\lVert (QMQ-c)\phi \rVert \le \lVert (M-c)\phi \rVert \). The second bound follows from \(\lVert (M-c)\phi \rVert \le \lVert (M-c)\psi \rVert + \lVert M-c\rVert \, \lVert \phi -\psi \rVert \).
For real numbers \(\epsilon \), \(a\) and \(s\), let \(b_0=0\) and \(b_{m+1}=(\epsilon +ab_m)^s\). The source takes \(a=2\) and \(s=1-1/h\), so that \(b_1=\epsilon ^{1-1/h}\) [ 239 , lines 696–704 ] . This scalar sequence is considered independently of the projection hierarchy.
For \(\epsilon \ge 0\), \(a\ge 0\) and real \(s\), every \(b_m\) is nonnegative.
By induction on \(m\): \(b_0=0\), and if \(b_m\ge 0\) then \(\epsilon +ab_m\ge 0\), so its real power \(b_{m+1}\) is nonnegative.
Let \(0{\lt}\epsilon \le 1\), \(a\ge 0\) and \(0\le s\le 1\). Then for every \(m\),
and for \(a=2\), \(b_m\le (2^m-1)\epsilon ^{s^m}\).
By induction on \(m\). With \(C=\sum _{j{\lt}m}a^j\), the inequality \(\epsilon \le \epsilon ^{s^m}\) gives \(\epsilon +ab_m\le (1+aC)\epsilon ^{s^m}\), and \(1+aC=\sum _{j{\lt}m+1}a^j\ge 1\), so \((1+aC)^s\le 1+aC\) and \(b_{m+1}\le (1+aC)\epsilon ^{s^{m+1}}\).
For every natural number \(h\), \(0\le 1-1/h\le 1\), with \(1/0\) read as \(0\). For \(h\ge 2\), \((1-1/h)^h\ge 1/4\).
The cases \(h=2,3\) are direct. For \(h=n+1\ge 4\), \(1-1/h\ge e^{-1/n}\) because \(1+1/n\le e^{1/n}\), so \((1-1/h)^n\ge e^{-1}\ge 1/3\), and \(1-1/h\ge 3/4\).
Let \(0{\lt}\epsilon \le 1\), \(h\) a natural number, \(a=2\) and \(s=1-1/h\). Then \(b_h\le (2^h-1)\epsilon ^{(1-1/h)^h}\). For \(\epsilon =e^{-t}\) with \(t\ge 0\),
which for \(N=2^h\) and \(t=l_{\mathrm{proj}}/l_{\mathrm{fac}}\) is the bound \(N\exp [-(l_{\mathrm{proj}}/l_{\mathrm{fac}})(1-1/h)^h]\) of [ 239 , lines 705–707 ] . For \(h\ge 2\), \(b_h\le (2^h-1)\epsilon ^{1/4}\). The source’s further estimate \(N\exp (-l_{\mathrm{proj}}/(e\, l_{\mathrm{fac}}))\) uses the limit \((1-1/h)^h\to 1/e\), which is approached from below, so here the constant \(1/4\) takes its place [ 138 ] .
32.7 All levels of Hastings’ hierarchical projection
The hierarchy of [ 239 , lines 668–707 ] projects, at level \(m+1\), onto the window of the pair operator joining the two halves of a block of \(2^{m+1}\) supersites, inside the range of the product of the two projections of level \(m\). This section bounds the deficiency of every level. Each deficiency is measured against the ground state \(\Psi _a\) itself, rather than against the normalized vector of the level below as in the source; this replaces the printed recurrence by a recurrence for squared deficiencies, with coefficient \(3^m-1\) in place of \(2^m-1\) [ 138 ] . The two halves of a block act on disjoint supersites, so their projections commute; here this is a hypothesis.
Let \(P,P'\) be commuting orthogonal projections, \(M\) Hermitian, \(c\) real, \(x{\gt}0\), and \(\psi \) a unit vector with \(\lVert (M-c)\psi \rVert \le \epsilon \), \(\lVert \psi -P\psi \rVert \le D\) and \(\lVert \psi -P'\psi \rVert \le D\). Let \(R\) be the window of \(M\) at \(c\) with radius \(x\) inside the range of \(PP'\). Then
Put \(Q=PP'\) and \(\psi '=Q\psi /\lVert Q\psi \rVert \). The vectors \(\psi -P\psi \) and \(P(\psi -P'\psi )\) are orthogonal, so \(\lVert \psi -Q\psi \rVert ^2\le 2D^2\). The vector \(\psi '\) lies in the range of \(Q\) and \(\lVert \psi '-\psi \rVert \le 2D\), so the window moves it by at most \((\epsilon +2\lVert M-c\rVert D)/x\). Since \(RQ=R\), the nested projection bound combines the two.
For real \(\epsilon \), \(K\) and radii \(x_0,x_1,\dots \), put \(D_0=0\) and
Let \(P_k^0=1\) for every block \(k\), and let \(P_k^{m+1}\) be the window of the Hermitian pair operator \(M_k^m\) at \(c_k^m\) with radius \(x_m{\gt}0\) inside the range of \(P_{2k}^mP_{2k+1}^m\). Suppose \(P_{2k}^m\) and \(P_{2k+1}^m\) commute, and that \(\lVert (M_k^m-c_k^m)\psi \rVert \le \epsilon \) and \(\lVert M_k^m-c_k^m\rVert \le K\) for all \(m\) and \(k\), where \(\psi \) is a unit vector. Then every \(P_k^m\) is an orthogonal projection and \(\lVert \psi -P_k^m\psi \rVert \le D_m\). In the source \(\psi =\Psi _a\) and \(c_k^m=\langle \Psi _a,M_k^m\Psi _a\rangle \) [ 239 , lines 668–707 ] .
Induction on \(m\), applying Theorem 32.7.1 to the two halves of each block and using that the right side is increasing in \(\lVert M-c\rVert \).
For real \(e\) and \(s\), put \(\beta _0=0\) and \(\beta _{m+1}=\bigl(2\beta _m^2+(e+2\beta _m)^{2s}\bigr)^{1/2}\).
Let \(K{\gt}0\) and \(e{\gt}0\). With \(\epsilon =Ke\) and the radii \(x_m=K(e+2\beta _m)^{1-s}\) of [ 239 , lines 676–681 ] , \(D_m\le \beta _m\) and \(0\le \beta _m\) for every \(m\).
By induction, \((\epsilon +2KD_m)/x_m\le (e+2\beta _m)/(e+2\beta _m)^{1-s} =(e+2\beta _m)^s\).
For \(0{\lt}e\le 1\) and \(0\le s\le 1\), \(\beta _m\le (3^m-1)e^{s^m}\). For the source’s \(s=1-1/h\), \(N=2^h\) supersites and \(h\ge 2\), the majorant of the last level satisfies \(\beta _h\le (3^h-1)e^{1/4}\), in place of the printed \(N\exp [-(l_{\mathrm{proj}}/l_{\mathrm{fac}})(1-1/h)^h]\) for \(e=\exp (-l_{\mathrm{proj}}/l_{\mathrm{fac}})\) [ 239 , lines 705–707 ] .
Write \(\beta _m\le C_me^{s^m}\) with \(C_m=3^m-1\). Since \(e\le e^{s^m}\le e^{s^{m+1}}\) and \(y^s\le y\) for \(y\ge 1\), \((e+2\beta _m)^s\le (1+2C_m)e^{s^{m+1}}\), and \(2C_m^2+(1+2C_m)^2\le (3C_m+2)^2=C_{m+1}^2\). For \(s=1-1/h\) and \(m=h\), \((1-1/h)^h\ge 1/4\) by Theorem 32.6.17, and \(e^{(1-1/h)^h}\le e^{1/4}\) since \(e\le 1\).
32.7.1 The supersite grouping
The source groups the sites into supersites of \(2l_{\mathrm{proj}}\) sites, so that each \(M_i\) acts on at most two neighbouring supersites, and lets \(M_{k,k+1}\) be the sum of the \(M_i\) acting on supersites \(k\) and \(k+1\) [ 239 , lines 609–628 ] . Block \(k\) of level \(m\) consists of the sites whose supersite has quotient \(k\) by \(2^m\).
If an operator \(A\) acts on a set of sites \(S\), so does \(f(A)\) for every real function \(f\). In particular the window of \(M\), and the window of \(M\) inside the range of \(Q\), act on \(S\) when \(M\) and \(Q\) do.
The operators acting on \(S\) form a closed \(*\)-subalgebra containing \(A\), and \(f(A)\) lies in every such subalgebra.
Given the supersite \(\sigma (i)\in \mathbb N\) of each site \(i\), block \(k\) of level \(m\) is the set of sites \(i\) with \(\lfloor \sigma (i)/2^m\rfloor =k\).
Blocks \(2k\) and \(2k+1\) of level \(m\) are disjoint and contained in block \(k\) of level \(m+1\).
\(\lfloor \sigma /2^{m+1}\rfloor =\lfloor \lfloor \sigma /2^m\rfloor /2\rfloor \).
For operators \(M_i\) and reals \(c_i\), \(\lVert (\sum _iM_i-\sum _ic_i)\psi \rVert \le \sum _i\lVert (M_i-c_i)\psi \rVert \). With \(c_i=\langle \Psi _a,M_i\Psi _a\rangle \), this bounds the residual of \(M_{k,k+1}\) by eq. (low) for its summands [ 239 , lines 628–639 ] .
The triangle inequality.
32.8 The energy of the final state of Hastings’ hierarchical construction
At the end of the hierarchical construction [ 239 ] states that, once the deficiency \(b_h\) of the last level is small, some final state has energy close to that of the ground state, eq. (hchy) at lines 709–720; the source writes “one may show”. The step from the deficiency of the final projection to the energy is the following. Here the deficiency is measured in norm, as in Section 32.6, and the state produced is the normalized image of the ground state, which lies in the span of the final states. The second part of this section proves eq. (hchy) for every final state, from the spectral windows rather than from the deficiency, first for the links of the open chain and then, with one further window, for the periodic chain [ 151 ] .
For unit vectors \(\phi \), \(\psi \) and every operator \(M\), \(|\operatorname {Re}\langle \phi ,M\phi \rangle -\operatorname {Re}\langle \psi ,M\psi \rangle | \le 2\| M\| \, \| \phi -\psi \| \).
\(\langle \phi ,M\phi \rangle -\langle \psi ,M\psi \rangle =\langle \phi -\psi ,M\phi \rangle +\langle \psi ,M(\phi -\psi )\rangle \), and each term has modulus at most \(\| M\| \, \| \phi -\psi \| \).
If the orthogonal projection \(Q\) moves the unit vector \(\psi \) by at most \(b\), then \(\| Q\psi /\| Q\psi \| -\psi \| \le 2b\).
Normalizing a vector \(v\) moves it by \(|1-\| v\| |\le \| v-\psi \| \), so the normalized image is within \(2\| Q\psi -\psi \| \) of \(\psi \).
Let \(Q\) be an orthogonal projection that moves the unit vector \(\Psi \) by at most \(b{\lt}1\). Then some unit vector \(\phi \) with \(Q\phi =\phi \) and \(\| \phi -\Psi \| \le 2b\) satisfies, for every operator \(M\),
For \(M=\sum _{i{\lt}N}M_i\) with \(\| M_i\| \le J\) the error is at most \(4NJb\). With \(Q\) the projection onto the final states and \(b=b_h\) this is eq. (hchy) of [ 239 , lines 709–720 ] , with the explicit constant \(4NJb_h\), for a unit vector in the span of the final states \(\alpha ^h\) rather than one of them [ 151 ] .
Every final state
Let \(A\) be Hermitian, \(c\) real and \(x\ge 0\), and let \(\phi \) be kept by the projection onto the eigenvectors of \(A\) with eigenvalue within \(x\) of \(c\). Then \(|\operatorname {Re}\langle \phi ,A\phi \rangle -c\| \phi \| ^2|\le x\| \phi \| ^2\). If instead \(\phi \) is kept by the window of a Hermitian \(M\) at \(c\) and \(x\) inside the range of an orthogonal projection \(Q\), then \(Q\phi =\phi \) and \(|\operatorname {Re}\langle \phi ,M\phi \rangle -c\| \phi \| ^2|\le x\| \phi \| ^2\).
On the range of the window, \(\| (A-c)\phi \| \le x\| \phi \| \), and \(\langle \phi ,A\phi \rangle -c\| \phi \| ^2=\langle \phi ,(A-c)\phi \rangle \). For the second statement, the window inside the range of \(Q\) is \(Q\) times the window of \(QMQ\), so \(Q\phi =\phi \), the window of \(QMQ\) keeps \(\phi \), and \(\langle \phi ,QMQ\phi \rangle =\langle \phi ,M\phi \rangle \).
For a family \(f_k^m\) indexed by levels \(m\) and blocks \(k\), set \(T_k^0(f)=0\) and \(T_k^{m+1}(f)=f_k^m+T_{2k}^m(f)+T_{2k+1}^m(f)\). With \(f_k^m\) the operator joining the two halves of block \(k\) of level \(m+1\), \(T_0^h(f)\) is the sum of the \(2^h-1\) links of an open chain of \(2^h\) supersites.
In the setting of Theorem 32.7.3, with radii \(x_m\ge 0\), every \(P_k^m\) is an orthogonal projection, and every vector \(\phi \) with \(P_k^m\phi =\phi \) satisfies
where \(T_k^m(x)\) sums the radius \(x_{m'}\) of every joining operator of level \(m'\) below the block.
Induction on \(m\). A vector kept by \(P_k^{m+1}\) is kept by the window of \(M_k^m\) inside the range of \(Q=P_{2k}^mP_{2k+1}^m\), so Theorem 32.8.4 bounds the term of \(M_k^m\) and gives \(Q\phi =\phi \). Since the two halves commute, \(\phi \) is kept by \(P_{2k}^m\) and by \(P_{2k+1}^m\), and the induction hypothesis bounds the two sums below them.
In the setting of Theorem 32.8.6, let the centres be \(c_k^m=\operatorname {Re}\langle \Psi ,M_k^m\Psi \rangle \) for a unit vector \(\Psi \). Then every unit vector \(\phi \) with \(P_k^m\phi =\phi \) satisfies
With \(\Psi =\Psi _a\) and the final projection \(P_0^h\), every final state \(\alpha ^h\) satisfies eq. (hchy) of [ 239 , lines 709–720 ] , with the error \(\sum _{m{\lt}h}2^{h-1-m}x_m\), for the energy of the links of the open chain [ 151 ] .
The expectation in \(\Psi \) is additive in the operator, so \(T_k^m(c)=\operatorname {Re}\langle \Psi ,T_k^m(M)\Psi \rangle \); apply Theorem 32.8.6 with \(\| \phi \| =1\).
In the setting of Theorem 32.8.7, let \(H=T_0^h(M)\) satisfy \(H-E_0\ge \Delta E\, (1-P)\) for an orthogonal projection \(P\) and \(\Delta E{\gt}0\), and suppose \(\operatorname {Re}\langle \Psi ,H\Psi \rangle +T_0^h(x)\le E_0+\Delta E/2\). Then every orthonormal family of vectors kept by \(P_0^h\) has at most \(2\operatorname {Re}\operatorname {tr}P\) members. This is the count of [ 239 , lines 722–730 ] for the open chain.
The chain of supersites is periodic, and the link \((N,1)\) closing it into a ring joins the two halves of no block, so the windows of the \(h\) levels leave its energy uncontrolled. One further window, inside the final space, controls it [ 151 ] .
In the setting of Theorem 32.7.3, let \(W\) be Hermitian with \(\| (W-c_W)\psi \| \le \epsilon \) and \(\| W-c_W\| \le K\), and let \(R\) project onto the eigenvectors of \(P_0^hWP_0^h\) in the range of \(P_0^h\) with eigenvalue within \(x_h\) of \(c_W\). Then \(R\) is an orthogonal projection and \(\| \psi -R\psi \| \le D_{h+1}\).
The window \(R\) is the step of a further level whose halves are \(P_0^h\) and the identity. The identity commutes with \(P_0^h\) and does not move \(\psi \), so Theorem 32.7.1 with the deficiency \(D_h\) of \(P_0^h\) gives \(\| \psi -R\psi \| ^2\le D_{h+1}^2\).
In the setting of Theorem 32.8.7, let \(W\) be Hermitian, \(x_W\ge 0\), and let \(R\) project onto the eigenvectors of \(P_0^hWP_0^h\) in the range of \(P_0^h\) with eigenvalue within \(x_W\) of \(\operatorname {Re}\langle \Psi ,W\Psi \rangle \). Then every unit vector \(\phi \) with \(R\phi =\phi \) satisfies
With \(W=M_{N,1}\) and \(\Psi =\Psi _a\), the operator \(T_0^h(M)+W\) is \(\sum _iM_i\), and this is eq. (hchy) of [ 239 , lines 709–720 ] for every state kept by \(R\), a subspace of the source’s final space spanned by the \(\alpha ^h\) [ 151 ] .
In the setting of Theorem 32.8.10, let \(H=T_0^h(M)+W\) satisfy \(H-E_0\ge \Delta E\, (1-P)\) for an orthogonal projection \(P\) and \(\Delta E{\gt}0\), and suppose \(\operatorname {Re}\langle \Psi ,H\Psi \rangle +T_0^h(x)+x_W\le E_0+\Delta E/2\). Then every orthonormal family of vectors kept by \(R\) has at most \(2\operatorname {Re}\operatorname {tr}P\) members. This is the count of [ 239 , lines 722–730 ] under the gap of the periodic Hamiltonian.
32.9 The Gaussian time average in Hastings’ approximate ground-state projector
The last step of the proof of Lemma 1 of [ 240 ] replaces the ground-state projector by a Gaussian average of the evolution, and then replaces the Hamiltonian in that average by a nearby operator. This section states both estimates for finite complex matrices, with explicit constants in the parameter \(q\). Norms are operator norms.
Let \(q\) and \(\Delta E\) be real numbers and put \(h=\max (q/\Delta E^2,0)\) if \(\Delta E\ne 0\), and \(h=0\) if \(\Delta E=0\). For a complex square matrix \(H\) let
where \(\gamma _h\) is the centred Gaussian measure of variance \(h\), and \(\gamma _0\) is the point mass at zero. For \(q{\gt}0\) and \(\Delta E{\gt}0\) this is the operator \(P_q=\frac{\Delta E}{\sqrt{2\pi q}}\int e^{iHt}e^{-(t\Delta E)^2/2q}\, dt\) of [ 240 , lines 843–847 ] . The variance satisfies \(q\le h\, \Delta E^2\) whenever \(\Delta E\ne 0\), and \(\sqrt h=\sqrt{\max (q,0)}/\Delta E\) whenever \(\Delta E{\gt}0\).
Let \(H\) be Hermitian, let \(\Omega \) be a unit vector with \(H\Omega =0\), let \(\Delta E{\gt}0\), and suppose that \(H\ge \Delta E\, (1-|\Omega \rangle \langle \Omega |)\). Then for every real \(q\),
This is the estimate of \(\| P_q-P_0\| \) at [ 240 , line 848 ] , for the unique gapped ground state with energy zero. With the source’s choice \(q=(l/3)\Delta E/(2v)\) it gives only \(O(e^{-l/6\xi })\), not the printed \(O(e^{-l/3\xi })\); the same factor enters the two preceding estimates on the filtered pieces of the Hamiltonian and the margin of the spectral cut-offs. Taking \(q=(l/3)\Delta E/v\) recovers the printed rate, see [ 148 ] .
Let \((e_k)\) be an orthonormal eigenbasis of \(H\) with eigenvalues \(\lambda _k\). The Gaussian characteristic function gives \(\langle e_k,P_q(H)x\rangle =e^{-h\lambda _k^2/2}\langle e_k,x\rangle \). Each \(e_k\) is either a multiple of \(\Omega \) with \(\lambda _k=0\), or is orthogonal to \(\Omega \) with \(\lambda _k\ge \Delta E\). Hence the coefficients of \(P_q(H)x-\langle \Omega ,x\rangle \Omega \) are bounded by \(e^{-h\Delta E^2/2}\) times those of \(x\), and Parseval’s identity gives the bound \(e^{-h\Delta E^2/2}\| x\| \). Finally \(h\Delta E^2\ge q\).
For Hermitian matrices \(M\) and \(H\) and every real \(t\),
By the Duhamel formula, \(e^{-iMt}e^{iHt}-1=\int _0^te^{-iMs}\, i(H-M)\, e^{iHs}\, ds\). The integrand has norm \(\| M-H\| \), and multiplication by the unitary \(e^{iMt}\) preserves the norm.
For \(h,T\ge 0\),
where the right-hand side is interpreted as zero when \(h=0\).
If \(h=0\), the measure is concentrated at zero and the integral vanishes. For \(h{\gt}0\), write \(\varphi _h(t)=(2\pi h)^{-1/2}e^{-t^2/(2h)}\). Since \((-h\varphi _h)'(t)=t\varphi _h(t)\) and \(\varphi _h(t)\to 0\) as \(t\to \infty \), one has \(\int _T^\infty t\varphi _h(t)\, dt=h\varphi _h(T)\). Reflection gives the same contribution on \((-\infty ,-T)\), hence the asserted bound, with equality for \(h{\gt}0\).
For every \(h\ge 0\), \(\int |t|\, d\gamma _h(t)\le \sqrt{2h/\pi }\le \sqrt h\).
Specialize Theorem 32.9.4 to \(T=0\). Since \(|t|\) vanishes at zero, removing that point does not change the integral, even when \(h=0\). Finally, \(2\le \pi \) gives \(\sqrt{2h/\pi }\le \sqrt h\).
Let \(M\) and \(H\) be Hermitian, let \(\Delta E{\gt}0\), let \(q\) be real and write \(\sqrt q\) for \(\sqrt{\max (q,0)}\). Then
If moreover \(\Omega \) and \(H\) satisfy the hypotheses of Theorem 32.9.2, then
With \(M=M_L+M_B+M_R\) this is the first inequality of the display of [ 240 , lines 849–864 ] , before the evolution of \(M\) is written in the interaction picture.
32.10 The cut-offs in front of the Gaussian time average
After the Gaussian time average has been moved from the Hamiltonian to the nearby operator \(M=M_L+M_B+M_R\), the proof of Lemma 1 of [ 240 ] multiplies it on the right by the spectral cut-offs \(O_L\) and \(O_R\). Since the cut-offs nearly fix the ground state, this changes the distance to the ground-state projector only slightly. This section states the estimate for finite complex matrices with explicit constants. Norms are operator norms.
Let \(O\) be a self-adjoint matrix and \(\Omega \) a unit vector. Then
Taking adjoints, the norm equals that of \((O-1)|\Omega \rangle \langle \Omega |\), which maps \(x\) to \(\langle \Omega ,x\rangle (O\Omega -\Omega )\). The Cauchy–Schwarz inequality bounds its norm by \(\| x\| \, \| O\Omega -\Omega \| \).
Let \(H\), \(\Omega \) and \(\Delta E{\gt}0\) satisfy the hypotheses of Theorem 32.9.2, and let \(M\) be Hermitian with \(\| M-H\| \le \delta \). Let \(O_L\) and \(O_R\) be commuting orthogonal projections with \(\| \Omega -O_LO_R\Omega \| \le \eta \). Then for every real \(q\),
The same bound holds with \(O_LO_R\) replaced by a single orthogonal projection. With \(M=M_L+M_B+M_R\), \(\delta \) of order \(\Delta E^{-1}J^2e^{-l/3\xi }\), and \(\eta \) of order \(e^{-l/6\xi }\) by the estimate on the spectral cut-offs, this is the estimate on [ 240 , lines 865–876 ] ; the term \(e^{-q/2}\) is the gap bound of [ 240 , line 848 ] .
Write \(P_0=|\Omega \rangle \langle \Omega |\) and \(O=O_LO_R\), an orthogonal projection since the factors commute. Then \(P_q(M)O-P_0=(P_q(M)-P_0)O+(P_0O-P_0)\). Since \(\| O\| \le 1\), Theorem 32.9.6 bounds the first term by \(e^{-q/2}+\sqrt q\, \delta /\Delta E\), and Theorem 32.10.1 bounds the second by \(\eta \).
32.11 The interaction picture in Hastings’ approximate ground-state projector
The last step of the proof of Lemma 1 of [ 240 ] writes the evolution generated by \(M_L+M_B+M_R\) as a time-ordered exponential of the interaction-picture boundary term, multiplied by the evolution generated by \(M_L+M_R\). This section states the ingredients for finite complex matrices \(A\) and \(B\), with \(A\) in the role of \(M_L+M_R\) and \(B\) in the role of \(M_B\). Norms are operator norms.
For complex square matrices \(A\) and \(B\) and a real number \(t\), put
Let \(A\) and \(B\) be complex square matrices. The path \(U\) is differentiable, \(U(0)=1\) and
for every real \(t\), and every differentiable path \(W\) with \(W(0)=1\) and \(W'(t)=W(t)\, iB(t)\) for all \(t\) equals \(U\). Further,
and if \(A\) and \(B\) are Hermitian then \(U(t)\) is unitary. Thus \(U(t)\) is the time-ordered exponential of \(i\int _0^tB(t')\, dt'\) of [ 240 , lines 853–864 ] , with later times to the right.
Differentiating the product gives \(U'(t)=e^{i(A+B)t}\, iB\, e^{-iAt}=U(t)\, iB(t)\), and \(U(0)=1\). Put \(V(t)=e^{iAt}e^{-i(A+B)t}\), so that \(V'(t)=-iB(t)V(t)\) and \(V(t)U(t)=1\). For a solution \(W\), the derivative of \(W(t)V(t)\) is \(W(t)\, iB(t)V(t)-W(t)\, iB(t)V(t)=0\), so \(W(t)V(t)=W(0)V(0)=1\) and \(W(t)=W(t)V(t)U(t)=U(t)\). The factorization follows from \(e^{-iAt}e^{iAt}=1\), and \(U(t)\) is a product of two unitaries when \(A\) and \(A+B\) are Hermitian.
Let \(K\) be a Hermitian matrix and \(X\) a matrix of the same size. For every real \(t\),
In [ 240 , lines 878–880 ] this is applied with \(K=M_L+M_R\) and \(X=O_LO_R\).
The derivative of \(s\mapsto e^{iKs}X\) is \(e^{iKs}\, iKX\), whose norm is \(\| KX\| \) because \(e^{iKs}\) is unitary. The mean value inequality on the interval between \(0\) and \(t\) gives the bound.
For a variance \(h\geq 0\), let \(\gamma _h\) be the centered Gaussian measure of variance \(h\) on the real line, and put
For \(q{\gt}0\) and \(\Delta E{\gt}0\), with \(h=q/\Delta E^2\), the density of \(\gamma _h\) is \((\Delta E/\sqrt{2\pi q})e^{-(t\Delta E)^2/2q}\), and \(P_B\) is the operator of [ 240 , lines 891–897 ] .
If \(A\) and \(B\) are Hermitian, then \(\| P_B\| \leq 1\) for every variance.
Each \(U(t)\) is unitary, and \(\gamma _h\) is a probability measure.
Let \(A\) and \(B\) be Hermitian, let \(X\) and \(P_0\) be matrices of the same size, and let \(h\geq 0\). Then
With \(A=M_L+M_R\), \(B=M_B\), \(X=O_LO_R\) and \(h=q/\Delta E^2\), this is the step from the estimate with \(e^{i(M_L+M_B+M_R)t}\) to eq. (last) of [ 240 , lines 881–890 ] : if \(\| (M_L+M_R)O_LO_R\| \leq \delta \) and \(\| \int e^{i(M_L+M_B+M_R)t}\, d\gamma _h(t)\, O_LO_R-P_0\| \leq \varepsilon \), then \(\| P_BO_LO_R-P_0\| \leq \varepsilon +\sqrt h\, \delta \).
By the factorization, \(e^{i(A+B)t}X-U(t)X=U(t)\bigl(e^{iAt}X-X\bigr)\), whose norm is at most \(|t|\, \| AX\| \) by the cut-off commutation estimate, since \(U(t)\) is unitary. Integrating against \(\gamma _h\) and using the first absolute moment bound gives \(\bigl\| \int e^{i(A+B)t}\, d\gamma _h\, X-P_BX\bigr\| \leq \sqrt h\, \| AX\| \), and the triangle inequality concludes.
32.12 The bootstrap lemma in Hastings’ area law
The bootstrap lemma of [ 240 ] turns an overlap of the ground state with a mixture of states of bounded Schmidt rank into a bound on the entanglement entropy across a cut. Its proof has three steps: the approximate factorization of the ground-state projector maps the mixture to mixtures of slightly larger Schmidt rank with large overlap, the overlap at each scale bounds the Schmidt coefficients beyond that rank, and the entropy of Schmidt coefficients obeying these geometric tail bounds is maximized. This section states the three steps with explicit constants. The lemma for the ground state of a gapped chain combines them with the factorization of [ 240 , Lemma 1 ] .
Let \(p\) be a probability vector on a finite set and let \(w\) be positive weights with \(\sum _zw_z\le 1\). Then
For \(p_z{\gt}0\), \(p_z\ln (w_z/p_z)\le w_z-p_z\) since \(\ln t\le t-1\); for \(p_z=0\) the term is \(0\le w_z\). Summing gives \(H(p)+\sum _zp_z\ln w_z\le \sum _zw_z-1\le 0\).
Let \(p_0,\dots ,p_{N-1}\) be a probability vector and let \(K\ge 1\), \(D\ge 2\) and \(0{\lt}x{\lt}1\). If for every \(n\ge 0\) the mass of the entries of index at least \(KD^{2n}\) is at most \(x^{n+1}\), then
In particular, with \(x=e^{-2/\xi '}\) and \(K=kD^{2m+2}\) for \(k\ge 1\), a natural number \(m\) and \(\xi '{\gt}0\),
This is the maximization in the proof of [ 240 , Lemma 2 ] , lines 357–390, where it is applied to the squared Schmidt coefficients in decreasing order.
The source finds the maximum by saturating every constraint; here the same value comes from Theorem 32.12.1 against the saturating weights. On the block of indices in \([KD^{2(n-1)},KD^{2n})\), of size \(KD^{2(n-1)}(D^2-1)\), and on \([0,K)\) for \(n=0\), put the weight \((1-x)x^n\) divided by the size of the block. The weights of the indices below \(KD^{2N}\) add up to \(1-x^{N+1}\), so the weights of the indices below \(N\le KD^{2N}\) add up to at most \(1\), and \(-\ln \) of a weight is \(\ln K-\ln (1-x)\) plus one increment for each threshold \(KD^{2i}\le a\) below the index \(a\). Averaging against \(p\) and bounding the mass beyond each threshold by \(x^{i+1}\) gives the first bound. With \(x=e^{-2/\xi '}\) one has \(x/(1-x)\le \xi '/2\), which gives the three estimates of the source, lines 386–389.
For a unit vector \(\Psi \) and any vector \(v\),
Expand the square on the right and use \(\| \Psi \| =1\).
Let \(\Psi _0\) be a unit vector and \(\rho =\sum _iw_i|\psi _i\rangle \langle \psi _i|\) a mixture of unit trace with \(\langle \Psi _0,\rho \Psi _0\rangle =P{\gt}0\). If an operator \(O\) satisfies \(\| O-|\Psi _0\rangle \langle \Psi _0|\| \le \epsilon \), then
and \(\operatorname {tr}(O\rho O^\dagger ){\gt}0\) when \(\epsilon {\lt}\sqrt P\). This is equation hg of [ 240 , Lemma 2 ] , lines 317–331, with \(\epsilon =C_1(\xi )e^{-m/\xi '}\).
Write \(a=\langle \Psi _0,O\rho O^\dagger \Psi _0\rangle \) and \(b=\operatorname {tr}((1-P_0)O\rho O^\dagger )\), where \(P_0=|\Psi _0\rangle \langle \Psi _0|\). The weighted Minkowski inequality gives \(a\ge (\sqrt P-\epsilon )^2\) for \(\epsilon \le \sqrt P\), and \((1-P_0)O=(1-P_0)(O-P_0)\) gives \(b\le \epsilon ^2\). For \(\epsilon {\lt}\sqrt P\) the trace \(a+b\) is positive and \(b/(a+b)\le \epsilon ^2/((\sqrt P-\epsilon )^2+\epsilon ^2)\le 2\epsilon ^2/P\), since \((\sqrt P-\epsilon )^2+\epsilon ^2=2(\epsilon -\sqrt P/2)^2+P/2\). For \(\epsilon \ge \sqrt P\) the bound is trivial.
Let \(\varphi \) be a bipartite vector with coefficient matrix \(C_\varphi \), and let \(\rho _\varphi =C_\varphi C_\varphi ^\dagger \) be its unnormalized reduced operator on the first factor, of trace \(\| \varphi \| ^2\). For a vector \(\psi \) of Schmidt rank at most \(K\), with \(K\) less than the dimension of the first factor, \(|\langle \varphi ,\psi \rangle |^2\le \| \psi \| ^2\, \| \rho _\varphi \| _{(K)}\), where \(\| \cdot \| _{(K)}\) is the sum of the \(K\) largest eigenvalues.
For \(\psi =0\) both sides vanish. Otherwise, normalizing \(\psi \) reduces it to the variational characterization of the maximal overlap with states of Schmidt rank at most \(K\) by the Ky Fan norm.
Let \(\Psi \) be a unit bipartite vector with reduced density matrix \(\rho _\Psi \), let \(K\) be a natural number and \(\eta \ge 0\), and let \(w_1,\dots ,w_r\ge 0\) and \(\psi _1,\dots ,\psi _r\) be vectors of Schmidt rank at most \(K\) with \(W=\sum _iw_i\| \psi _i\| ^2{\gt}0\). If \((1-\eta )W\le \sum _iw_i|\langle \Psi ,\psi _i\rangle |^2\), then \(1-\| \rho _\Psi \| _{(K)}\le \eta \).
If \(K\) is at least the dimension of the first factor, then \(\| \rho _\Psi \| _{(K)}=\operatorname {tr}\rho _\Psi =1\). Otherwise Theorem 32.12.5 gives \(|\langle \Psi ,\psi _i\rangle |^2\le \| \psi _i\| ^2\| \rho _\Psi \| _{(K)}\) for each \(i\), so \((1-\eta )W\le W\| \rho _\Psi \| _{(K)}\), and dividing by \(W\) gives the bound.
For finite-dimensional spaces \(\mathcal H_1\) and \(\mathcal H_2\), an operator of the form \(X\otimes 1\) or \(1\otimes X\) does not increase the rank of the coefficient matrix of a vector of \(\mathcal H_1\otimes \mathcal H_2\). On the chain of \(L\) sites with local dimension \(q\ge 1\), consider the cut after \(k\) sites and the Schmidt rank of a vector across it. Writing the chain as the product of the first \(k\) sites and the remaining sites, the coefficient matrix of \(X\psi \) across the cut is the coefficient matrix of the vector \(X\psi \) in these two factors, with \(X\) written as an operator on the product. Hence an operator supported on the first \(k\) sites, or on the sites after them, does not increase the Schmidt rank. An operator supported on the first \(k+m\) sites, with \(k+m\le L\), increases it by a factor of at most \(q^{2m}\). For an operator supported on the \(2m\) sites around the cut, with \(D=q\), this is the bound \(K\mapsto KD^{2m}\) on the Schmidt rank in the proof of [ 240 , Lemma 2 ] , lines 321–322.
Write \(C_\psi \) for the coefficient matrix of \(\psi \) in \(\mathcal H_1\otimes \mathcal H_2\). Then
and multiplying a matrix on one side does not increase its rank. For the second statement write \(r_k(\psi )\) for the Schmidt rank of \(\psi \) across the cut after \(k\) sites. Moving a cut by \(m\) sites changes the Schmidt rank by a factor of at most \(q^m\), and across the cut after \(k+m\) sites the operator \(X\) acts on one side, so
For \(\xi '{\gt}0\) and a natural number \(D\ge 2\), put
as in [ 240 , Lemma 2 ] , lines 304–309.
Let \(\Psi \) be a unit bipartite vector with reduced density matrix \(\rho _\Psi \), let \(k\) and \(D\) be natural numbers, and let \(\xi '{\gt}0\) and \(c\ge 1\). Suppose that at every scale \(s\) with \(ce^{-2s/\xi '}{\lt}1\) some mixture of states of Schmidt rank at most \(kD^{2s}\), of positive total weight, has normalized overlap with \(\Psi \) at least \(1-ce^{-2s/\xi '}\). Then the eigenvalues \(p_1\ge p_2\ge \dots \) of \(\rho _\Psi \) form a probability vector, and there is a natural number \(m'\) with \(m'{\lt}\tfrac {\xi ’}{2}\ln c+1\) such that for every \(n\ge 0\)
These are equations constraint and constraint2 in the proof of [ 240 , Lemma 2 ] , lines 333–356.
Since \(\rho _\Psi =C_\Psi C_\Psi ^\dagger \) is positive semidefinite with \(\operatorname {tr}\rho _\Psi =\lVert \Psi \rVert ^2=1\), its eigenvalues are nonnegative and, by Theorem 32.12.5 with no eigenvalue removed, sum to \(1\). At each such scale Theorem 32.12.6 bounds the mass of the Schmidt coefficients beyond the first \(kD^{2s}\) by \(ce^{-2s/\xi '}\), which is equation constraint. Let \(m'\) be the least natural number with \(ce^{-2m'/\xi '}\le 1\); then \(2m'{\lt}\xi '\ln c+2\), and the scale \(m'+1+n\) satisfies \(ce^{-2(m'+1+n)/\xi '}\le e^{-2(n+1)/\xi '}{\lt}1\).
Let \(\Psi \) be a unit bipartite vector with reduced density matrix \(\rho _\Psi \), and let \(k\ge 1\), \(D\ge 2\), \(\xi '{\gt}0\) and \(c\ge 1\). Suppose that at every scale \(s\) with \(ce^{-2s/\xi '}{\lt}1\) some mixture of states of Schmidt rank at most \(kD^{2s}\), of positive total weight, has normalized overlap with \(\Psi \) at least \(1-ce^{-2s/\xi '}\). Then
This is a stage of [ 240 , Lemma 2 ] : it takes the mixtures at every scale as a hypothesis, and with \(c=2C_1(\xi )^2/P\) its conclusion is equation bootstrap.
For vectors \(x,y\) on the chain of \(L\) sites and a cut \(j\), write \(\hat x,\hat y\) for their coordinates in the product of the first \(j\) sites and the remaining sites. Then \(\langle \hat x,\hat y\rangle =\langle x,y\rangle \).
The coordinates across the cut are a relabelling of the configurations of the chain.
For a vector \(\Omega \) on the chain of \(L\) sites and a cut \(j\), the partial trace of \(|\Omega \rangle \langle \Omega |\) over the sites after the first \(j\), an unnormalized reduced operator of trace \(\| \Omega \| ^2\), is \(C_\Omega C_\Omega ^\dagger \) for the coefficient matrix \(C_\Omega \) across the cut, so its entropy is the entropy of \(C_\Omega C_\Omega ^\dagger \).
The partial trace over the second factor of \(|\Omega \rangle \langle \Omega |\) is \(C_\Omega C_\Omega ^\dagger \) in the cut coordinates, and the first \(j\) sites are the interval \(\{ 0,\dots ,j-1\} \).
On the chain of \(L\) sites with local dimension \(q\ge 1\), if \(L{\lt}j+s\) then every vector has Schmidt rank at most \(q^{2s}\) across the cut after \(j\) sites.
If \(j\le L\), moving the cut to the end of the chain costs a factor at most \(q^{L-j}\le q^{2s}\), and the cut at the end has Schmidt rank at most one. If \(j{\gt}L\), the cut already contains the whole chain.
Let \(\Omega \) be a unit vector on the chain of \(L\) sites with local dimension \(q\ge 1\), let \(j\) be a cut, and let \(\xi '\) and \(C_1\) be real. Suppose that for every natural number \(m\) with \(j+m\le L\) there are operators \(O_B\) supported on the first \(j+m\) sites, \(O_L\) supported on the first \(j\) sites and \(O_R\) supported on the sites after them, with \(\| O_BO_LO_R-|\Omega \rangle \langle \Omega |\| \le C_1e^{-m/\xi '}\). Let \(w_1,\dots ,w_r\ge 0\) and \(\psi _1,\dots ,\psi _r\) be vectors of Schmidt rank at most \(k\ge 1\) across the cut, with \(\sum _iw_i\| \psi _i\| ^2=1\) and \(P=\sum _iw_i|\langle \psi _i,\Omega \rangle |^2{\gt}0\). If \(s\) is a natural number with \(c\, e^{-2s/\xi '}{\lt}1\), where \(c=2C_1^2/P\), then some mixture of vectors of Schmidt rank at most \(kq^{2s}\) across the cut, of positive total weight, has normalized overlap at least \(1-c\, e^{-2s/\xi '}\) with \(\Omega \). This is equation hg in the proof of [ 240 , Lemma 2 ] , lines 315–331. No sign is required of \(\xi '\): for \(\xi '=0\) the exponents \(-m/\xi '\) and \(-2s/\xi '\) are read as \(0\). The lemma is applied with \(\xi '{\gt}0\) in Theorem 32.12.15.
When \(j+s\le L\), the operator \(O=O_BO_LO_R\) maps each \(\psi _i\) to a vector of Schmidt rank at most \(kq^{2s}\) by Theorem 32.12.7, and Theorem 32.12.4 with \(\epsilon =C_1e^{-s/\xi '}{\lt}\sqrt P\) gives the mixture \(\sum _iw_i|O\psi _i\rangle \langle O\psi _i|\) the stated overlap. When \(j+s{\gt}L\), the vector \(\Omega \) itself serves, by Theorem 32.12.13.
Let \(\Omega \) be a unit vector on the chain of \(L\) sites with local dimension \(q\ge 2\), let \(j\) be a cut, and let \(\xi '{\gt}0\) and \(C_1\ge 1\). Suppose that for every natural number \(m\) with \(j+m\le L\) there are operators \(O_B\) supported on the first \(j+m\) sites, \(O_L\) supported on the first \(j\) sites and \(O_R\) supported on the sites after them, with
Let \(w_1,\dots ,w_r\ge 0\) and let \(\psi _1,\dots ,\psi _r\) be vectors of Schmidt rank at most \(k\ge 1\) across the cut after \(j\) sites, with \(\sum _iw_i\| \psi _i\| ^2=1\) and \(P=\sum _iw_i|\langle \psi _i,\Omega \rangle |^2{\gt}0\). Then the entropy \(S_j\) of the reduced density matrix of \(\Omega \) on the first \(j\) sites satisfies
This is a stage of [ 240 , Lemma 2 ] : the factorization of Lemma 1 is a hypothesis, and the Hamiltonian does not enter.
Let \(\Omega \) be a unit vector on the chain of \(L\) sites with local dimension \(q\ge 2\), let \(j\) be a cut, and let \(\xi '{\gt}0\) and \(C_1\ge 1\). Suppose that at the cut \(j\) and every length \(m\) there is an approximate factorization of \(|\Omega \rangle \langle \Omega |\) with error \(C_1e^{-m/\xi '}\). Then the conclusion of Theorem 32.12.15 holds for every mixture as there.
An approximate factorization at the cut \(j\) and length \(m\) has \(O_B\) supported on the sites \(j-m,\dots ,j+m-1\), hence on the first \(j+m\) sites, so it supplies the hypothesis of Theorem 32.12.15.
Let \(R\) be a range. There is a constant \(K\ge 1\), depending only on \(R\), with the following property. Let \(H\) be a Hamiltonian on a chain of \(L\) sites with local dimension \(D\), a sum of terms of range \(R\) and norm at most \(J\ge 0\), whose ground state \(\Psi _0\) is unique with gap \(\Delta E{\gt}0\) above the ground energy. Let \(\tilde\xi \) be the length of Theorem 32.4.14, and set \(\xi '=6\tilde\xi \) and \(C_1=K\tilde\xi ^3\). Let \(\rho \) be a density matrix of unit trace that is a mixture of pure states of Schmidt rank at most \(k\ge 1\) across the cut between sites \(j\) and \(j+1\), with \(\langle \Psi _0,\rho \Psi _0\rangle =P{\gt}0\). (The case \(k=0\) is vacuous: a vector of Schmidt rank zero vanishes, and then \(P=0\).) Then
where \(\rho ^0_{1,j}\) is the reduced density matrix of \(|\Psi _0\rangle \langle \Psi _0|\) on sites \(1,\dots ,j\). This is a quantitatively restricted form of [ 240 , Lemma 2 ] , lines 280–311, at the printed rate \(\xi '=6\xi \) for the length \(\xi =\tilde\xi \). The length \(\tilde\xi \) may stand in for the source’s \(\xi \): the source fixes \(\xi =\max (2v/\Delta E,\xi _C)\) only through some Lieb–Robinson pair, and a bound valid for \(|t|\le l/v\) stays valid for a larger velocity [ 240 , lines 96–112 ] , so raising the velocity of the chain estimate is a legitimate choice [ 137 ] . The constant \(C_1\) is bounded by a polynomial in the length, as the source requires, and is uniform in the chain length, local dimension, Hamiltonian, ground state, cut and mixture. The result inherits the remaining restriction of the formalized Lemma 1: the length \(\tilde\xi \) comes from the chain Lieb–Robinson estimate with its velocity raised by \(5/4\). That estimate is linear in the size of the support, as in the source, but its prefactor grows with \(J\) and its second operator acts on one site, so \(\tilde\xi \) is not shown to be the source’s \(\max (2v/\Delta E,\xi _C)\) for a Lieb–Robinson pair in the source’s sense [ 137 , 148 ] .
Theorem 32.4.14 gives a constant \(K_0\ge 0\) depending only on \(R\) and an approximate factorization of the ground-state projector with error \(K_0\tilde\xi ^3e^{-m/(6\tilde\xi )}\) at every cut and length; it also holds with \(K=\max (K_0,1)\), and then \(C_1=K\tilde\xi ^3\ge 1\) since \(\tilde\xi \ge 20\). For \(D\ge 2\), Theorem 32.12.16 gives the bound. For \(D=1\) the entropy vanishes, while the right side is nonnegative, with the convention \(\ln 0=0\) in the term \(\ln (D^2-1)\) of \(F(\xi ',D)\).
32.13 Rényi entropy under geometric tail constraints
The proof of [ 240 , Lemma 2 ] ends by extending the entropy bound of Theorem 32.12.2 to the Rényi entropies \(S_\alpha =(1-\alpha )^{-1}\ln \sum _ap_a^\alpha \) of sufficiently large order \(\alpha \). For \(\alpha \ge 1\) the Rényi entropy is at most the von Neumann entropy, so the content of the extension lies in the orders \(0{\lt}\alpha {\lt}1\) for which the relevant geometric series converges.
Let \(p_1,\dots ,p_c\ge 0\) and \(0\le \alpha \le 1\). Then
The map \(t\mapsto t^\alpha \) is concave on \([0,\infty )\), so by Jensen’s inequality the mean of the \(p_i^\alpha \) is at most the \(\alpha \)-th power of the mean of the \(p_i\). Multiplying by \(c\) gives the claim.
Let \(p_0,\dots ,p_{N-1}\) be a probability vector and let \(K\ge 1\), \(D\ge 2\) and \(x\ge 0\). Suppose that for every \(n\ge 0\) the mass of the entries of index at least \(KD^{2n}\) is at most \(x^{n+1}\). Let \(0{\lt}\alpha {\lt}1\) with \(r=(D^2)^{1-\alpha }x^\alpha {\lt}1\). Then
In particular, with \(x=e^{-2/\xi '}\) and \(K=kD^{2m+2}\) for \(k\ge 1\) and a natural number \(m\), the bound holds whenever \(e^{-2\alpha /\xi '}(D^2)^{1-\alpha }{\lt}1\), and reads
This is the extension to Rényi entropies in the proof of [ 240 , Lemma 2 ] , lines 392–410, with the convergence condition of line 406; it differs from the von Neumann bound only in the term replacing \(F(\xi ',D)\). The source’s maximization at lines 400–402 keeps a factor \((1-e^{-2/\xi '})^\alpha \) inside the logarithm, from the exact masses of the saturating blocks; bounding each block mass by its tail drops that factor, so the bound here is slightly weaker but of the same form.
Group the indices into the block \([0,K)\) and the blocks \([KD^{2(n-1)},KD^{2n})\) for \(n\ge 1\). The \(n\)-th block has at most \(KD^{2n}\) entries and mass at most \(x^n\): for \(n\ge 1\) by the tail constraint at \(n-1\), and for \(n=0\) because the total mass is \(1\). Theorem 32.13.1 bounds its contribution to \(\sum _ap_a^\alpha \) by \((KD^{2n})^{1-\alpha }x^{n\alpha }=K^{1-\alpha }r^n\), so \(\sum _ap_a^\alpha \le K^{1-\alpha }/(1-r)\). Taking logarithms and dividing by \(1-\alpha {\gt}0\) gives (42). With \(x=e^{-2/\xi '}\) one has \(x^\alpha =e^{-2\alpha /\xi '}\) and \(\ln K=\ln k+(2m+2)\ln D\).
Let \(\Psi \) be a unit vector of a bipartite space with reduced density matrix \(\rho \) on the first factor, and let \(k\ge 1\), \(D\ge 2\), \(\xi '{\gt}0\) and \(c\ge 1\). Suppose that at every scale \(s\) with \(ce^{-2s/\xi '}{\lt}1\) some mixture of pure states of Schmidt rank at most \(kD^{2s}\), of positive total weight, has normalized overlap with \(\Psi \) at least \(1-ce^{-2s/\xi '}\). Then for \(0{\lt}\alpha {\lt}1\) with \(r=(D^2)^{1-\alpha }e^{-2\alpha /\xi '}{\lt}1\),
This is the Rényi version of the stage of [ 240 , Lemma 2 ] for one cut, lines 333–356 and 392–410.
Let \(\Omega \) be a unit vector on a chain of \(L\) sites with local dimension \(D\ge 2\), and suppose that at the cut \(j\) and every length \(m\) there is an approximate factorization of \(|\Omega \rangle \langle \Omega |\) with error \(C_1e^{-m/\xi '}\), where \(\xi '{\gt}0\) and \(C_1\ge 1\). Let \(\rho \) be a mixture of unit trace of pure states of Schmidt rank at most \(k\ge 1\) across the cut with \(\langle \Omega ,\rho \Omega \rangle =P{\gt}0\). For \(0{\lt}\alpha {\lt}1\) with \(r=(D^2)^{1-\alpha }e^{-2\alpha /\xi '}{\lt}1\), the Rényi entropy of order \(\alpha \) of the first \(j\) sites satisfies
For \(D=1\) the Rényi entropy of order \(0{\lt}\alpha {\lt}1\) of the first \(j\) sites is at most \(0\).
As in Theorem 32.12.15, the factorization supplies at every scale \(s\) a mixture of Schmidt rank at most \(kD^{2s}\) with normalized overlap at least \(1-(2C_1^2/P)e^{-2s/\xi '}\), and \(2C_1^2/P\ge 1\). Theorem 32.13.3 with \(c=2C_1^2/P\) gives the bound. For \(D=1\) the reduced density matrix acts on a space of dimension one, so its single eigenvalue is \(1\) and \(\sum _a\lambda _a^\alpha =1\).
Let \(R\) be a range. There is a constant \(K\ge 1\), depending only on \(R\), with the following property. Let \(H\) be a Hamiltonian on a chain of \(L\) sites with local dimension \(D\), a sum of terms of range \(R\) and norm at most \(J\ge 0\), whose ground state \(\Psi _0\) is unique with gap \(\Delta E{\gt}0\). Let \(\tilde\xi \) be the length of Theorem 32.4.14, and set \(\xi '=6\tilde\xi \) and \(C_1=K\tilde\xi ^3\). Let \(0{\lt}\alpha {\lt}1\) with \(r=(D^2)^{1-\alpha }e^{-2\alpha /\xi '}{\lt}1\), and let \(\rho \) be a mixture of unit trace of pure states of Schmidt rank at most \(k\ge 1\) across the cut between sites \(j\) and \(j+1\) with \(\langle \Psi _0,\rho \Psi _0\rangle =P{\gt}0\). Then
This is the Rényi form of [ 240 , Lemma 2 ] , lines 392–410, with the same quantitative restriction as Theorem 32.12.17: the length \(\tilde\xi \) comes from the chain Lieb–Robinson estimate with raised velocity and is not shown to be the source’s \(\max (2v/\Delta E,\xi _C)\) for a Lieb–Robinson pair in the source’s sense [ 137 , 148 ] . It is proved here for orders \(0{\lt}\alpha {\lt}1\); order \(1\) is the von Neumann entropy of Theorem 32.12.17, and orders \(\alpha {\gt}1\) are Theorem 32.13.6 [ 166 ] .
Theorem 32.4.14 gives a constant \(K_0\ge 0\) and the factorization with error \(K_0\tilde\xi ^3e^{-m/(6\tilde\xi )}\) at every cut and length; it also holds with \(K=\max (K_0,1)\), and \(C_1=K\tilde\xi ^3\ge 1\) since \(\tilde\xi \ge 20\). Theorem 32.13.4 gives the bound for \(D\ge 2\) and the value at most \(0\) for \(D=1\), where the right side is nonnegative.
In the setting of Theorem 32.12.17, with the same constant \(K\), for every order \(\alpha {\gt}1\)
For \(\alpha {\gt}1\) the convergence condition of [ 240 , line 406 ] always holds, and the function replacing \(F(\xi ',D)\) is \(F(\xi ',D)\) itself; together with Theorem 32.13.5 this covers the orders of [ 240 , lines 392–410 ] .
For a density matrix with eigenvalues \(p_i\) and \(\alpha {\gt}1\), the concavity of the logarithm with weights \(p_i\) gives \(\ln \sum _ip_i^\alpha \ge (\alpha -1)\sum _ip_i\ln p_i\), so dividing by \(1-\alpha {\lt}0\) gives \(S_\alpha \le S\) (the bound of Rényi entropies of order above one in the companion quantum-channel volume [ 253 , “Rényi entropy of order above one” ] ). Theorem 32.12.17 bounds \(S\).
32.14 Decay of the Schmidt tail in Hastings’ area law
Let \(\Omega \) be a unit vector on a chain of \(L\) sites of local dimension \(q\ge 1\), let \(j\) be a cut and \(\rho \) the marginal of \(\Omega \) on the first \(j\) sites, with eigenvalues \(p_1\ge p_2\ge \dots \). Suppose that for every \(m\) with \(j+m\le L\) there are operators \(O_B\) supported on the first \(j+m\) sites, \(O_L\) supported on the first \(j\) sites and \(O_R\) supported on the remaining sites with \(\| O_BO_LO_R-|\Omega \rangle \langle \Omega |\| \le C_1e^{-m/\xi '}\). If \(k\ge 1\) and \(p_1+\cdots +p_k\ge 1/2\), then for every \(m\) and every \(n\ge kq^{2m}\),
The weight hypothesis holds for \(k\ge \exp (2S(\rho ))/2\) by Theorem 32.14.2; with that choice the conclusion is the constraint on the Schmidt tail with \(P=1/2\) in the section on matrix product states of [ 240 ] (Theorem 32.14.3).
Let \(E\) project the first \(j\) sites onto the eigenvectors of \(p_1,\dots ,p_k\) and put \(\psi =E\Omega \). Then \(\psi \) has Schmidt rank at most \(k\) across the cut, and \(\langle \psi ,\Omega \rangle =\| \psi \| ^2=p_1+\cdots +p_k=P\ge 1/2\). Theorem 32.12.14 applied to the state \(\psi /\| \psi \| \) gives, at every scale \(m\) with \((2C_1^2/P)e^{-2m/\xi '}{\lt}1\), a mixture of vectors of Schmidt rank at most \(kq^{2m}\) whose normalized overlap with \(\Omega \) is at least \(1-(2C_1^2/P)e^{-2m/\xi '}\), and Theorem 32.12.6 turns this into \(\sum _{a{\gt}kq^{2m}}p_a\le (2C_1^2/P)e^{-2m/\xi '}\le 4C_1^2e^{-2m/\xi '}\). At the remaining scales the right-hand side is at least one. The tail decreases as \(n\) grows.
Let \(\rho \) be a density matrix with eigenvalues \(p_1\ge p_2\ge \cdots \) and entropy \(S(\rho )\). If \(k\ge \exp (2S(\rho ))/2\), then \(p_1+\cdots +p_k\ge 1/2\). This is the bound that opens the section on matrix product states of [ 240 ] .
If \(k\) exceeds the dimension, the sum is the trace. Otherwise suppose \(p_1+\cdots +p_k{\lt}1/2\). Then \(p_k{\lt}1/(2k)\), so every \(p_a\) with \(a{\gt}k\) is below \(1/(2k)\), and the tail of weight more than \(1/2\) alone contributes more than \(\frac12\log (2k)\) to the entropy. Hence \(S(\rho ){\gt}\frac12\log (2k)\), that is \(k{\lt}\exp (2S(\rho ))/2\).
In the setting of Theorem 32.14.1, if \(k\ge \exp (2S(\rho ))/2\), then for every \(m\) and every \(n\ge kq^{2m}\) the eigenvalues of \(\rho \) beyond the \(n\) largest add up to at most \(4C_1^2e^{-2m/\xi '}\). This is the constraint on the Schmidt tail in the section on matrix product states of [ 240 ] , with \(k_0=\exp (2S(\rho ))/2\) and \(P=1/2\).
Let \(D{\gt}1\), \(\xi {\gt}0\), \(c\ge 0\) and \(k_0{\gt}0\), and let \(\tau \) be a real function on the natural numbers. Suppose that for all natural numbers \(m\) and \(k\) with \(k_0D^{2m}\le k\),
Then for every natural number \(k\ge k_0\),
Take \(\tau (k)\) to be the mass beyond the \(k\) largest squared Schmidt coefficients of the ground state across a cut, \(k_0=\exp (2S)/2\) for the entropy \(S\) of the cut, \(\xi \) the decay length written \(\xi '\) in the section on matrix product states of [ 240 ] , and \(c=4C_1^2\) for the constant \(C_1\) of the approximate factorization there. Then the hypothesis (43) is the consequence of the bootstrap lemma in that section, and (44) is the estimate that follows it, with two corrections: the constant \(e^{2/\xi }\) replaces the printed \(e^{1/\xi }\), and the sign of the exponent is restored [ 132 ] . The tail here starts after the \(k\)-th coefficient rather than at it: it is the tail that a matrix product state of bond dimension \(k\) discards. A bound for the sum starting at the \(k\)-th coefficient requires the separate initial-scale estimate described in that note.
Put \(x=\log (k/k_0)/\log D\ge 0\), let \(n\) be the integer part of \(x\) and \(m\) the integer part of \(n/2\). Then \(D^{2m}\le D^x=k/k_0\), so (43) applies at \(m\). Since \(2m\ge n-1{\gt}x-2\),
and \(e^{-x/\xi }=(k/k_0)^{-1/(\xi \log D)}\).
Let \(\psi \) be a unit vector on a chain of \(L\) sites, and let \(D{\gt}1\), \(\xi {\gt}0\), \(c\ge 0\) and \(k_0{\gt}0\). Suppose that at every internal cut the mass beyond the \(n\) largest eigenvalues of the left marginal is at most \(c\, e^{-2m/\xi }\) whenever \(k_0D^{2m}\le n\). Then for every natural number \(k'\ge k_0\), truncating every cut onto the \(k'\) leading eigenvectors of its left marginal gives an open-boundary matrix product state vector \(\Phi \), not necessarily normalized, of bond dimension at most \(k'\). If \(L{\lt}2\), then \(\Phi =\psi \). For \(L\ge 2\),
Here the mass beyond the \(n\) largest eigenvalues is the sum over the eigenvalues of index at least \(n+1\) in nonincreasing order, so the source’s sum from index \(k'\) is this mass at \(k'-1\). This is the conclusion of the section on matrix product states of [ 240 ] , whose error, stated without a measure, is the squared Hilbert-space norm of the unnormalized truncation [ 146 ] , with the constant of Theorem 32.14.4 [ 132 ] .
Sequential truncation onto the leading eigenvectors gives an open-boundary matrix product state of bond dimension at most \(k'\) with \(\lVert \Phi -\psi \rVert ^2\le \sum _k\varepsilon _k(k')\), where \(\varepsilon _k(k')\) is the mass beyond the \(k'\) largest eigenvalues at the cut \(k\). Theorem 32.14.4 bounds each \(\varepsilon _k(k')\), and there are \(L-1\) internal cuts.
Let \(\Omega \) be a unit vector on a chain of \(L\) sites with local dimension \(D\ge 1\), and let \(\xi '{\gt}0\). Suppose that the approximate factorization of Theorem 32.14.1 holds at every internal cut \(j\), and that the natural number \(k_0\) satisfies \(k_0\ge \exp (2S_j)/2\) for the entropy \(S_j\) of the first \(j\) sites at every internal cut \(j\). Then for every natural number \(k'\ge k_0\), truncating every cut onto the \(k'\) leading eigenvectors of its left marginal gives an open-boundary matrix product state \(\Phi \) of bond dimension at most \(k'\), without requiring \(\Phi \) to be normalized. For \(L{\lt}2\), there is no internal cut and \(\Phi =\Omega \). For \(D=1\) every left marginal is a \(1\times 1\) matrix of trace one, the truncation is exact and again \(\Phi =\Omega \); the exponent below, undefined at \(D=1\), is read there as \(0\), and the bound holds because its left side vanishes. The source assumes \(D\ge 2\). For \(L\ge 2\), the entropy hypothesis implies \(k_0{\gt}0\), and
The conditional construction in [ 240 ] explicitly assumes a uniform entropy bound. Its application to gapped ground states requires deriving both that bound and the factorization, with constants uniform in chain length, from the Hamiltonian assumptions [ 144 ] . Entropies and logarithms here use the natural base. For \(L\ge 2\), one may choose \(k_0=\lceil \exp (2\max _j S_j)/2\rceil \), with the maximum over internal cuts. The error is the squared Hilbert-space norm of the unnormalized vector [ 146 ] , and the constant is that of Theorem 32.14.4 [ 132 ] .
At every internal cut, Theorem 32.14.3 with \(k=k_0\) bounds the mass beyond the \(n\) largest eigenvalues by \(4C_1^2e^{-2m/\xi '}\) whenever \(k_0D^{2m}\le n\), since \(k_0\) is at least \(\exp (2S_j)/2\). Theorem 32.14.5 with \(c=4C_1^2\) then gives the bound. Without an internal cut, Theorems 31.6.6 and 31.6.7 give directly that \(\Phi \) is a matrix product state with \(\Phi =\Omega \).
The bound of Theorem 32.14.6 holds when, at every internal cut \(j\) and every length \(m\) with \(j+m\le L\), the approximate factorization is given in the form of the conclusion of Lemma 1 of [ 240 ] : commuting orthogonal projections \(O_L\) and \(O_R\) supported before and after the cut, and an operator \(O_B\) of norm at most one supported on the \(2m\) sites around the cut, cut short at the ends of the chain, with \(\lVert O_BO_LO_R-|\Omega \rangle \langle \Omega |\rVert \le C_1e^{-m/\xi '}\).
The operator \(O_B\) is supported on the first \(j+m\) sites, \(O_L\) on the first \(j\) sites and \(O_R\) on the remaining sites, which is the factorization of Theorem 32.14.6.
Let \(\Omega \) be a gapped ground vector of a finite-range Hamiltonian on a chain of \(L\) sites with local dimension \(D\ge 1\), and suppose that Lemma 1 of [ 240 ] holds around \(\Omega \) with constants \(\xi '{\gt}0\) and \(C_1\ge 1\), as in Theorem 32.22.4. Let \(S_{\max }\) be the bound of that theorem on the entropy of every initial segment, and \(k_0=\lceil \exp (2S_{\max })/2\rceil \). Then for every natural number \(k'\ge k_0\), truncating every cut onto the \(k'\) leading eigenvectors of its left marginal gives an open-boundary matrix product state \(\Phi \) of bond dimension at most \(k'\) with
For \(D=1\) the exponent is read as \(0\); the bound then holds because \(\Phi =\Omega \), as in Theorem 32.14.6. Lemma 1 is a hypothesis here, as in Theorem 32.22.4. With Lemma 1 supplied for the gapped ground state, this is the conclusion of the section on matrix product states of [ 240 ] , restricted in its quantitative bounds as Lemma 1 is: its length is not shown to be the source’s \(\max (2v/\Delta E,\xi _C)\), and the printed \(\xi '=6\xi \) is not reached; \(k_0\) inherits these through \(S_{\max }\) [ 137 , 148 ] . The error is measured in the squared norm [ 146 ] , and the constant is that of Theorem 32.14.4 [ 132 ] .
Let \(R\) be a range. There is a constant \(K\ge 0\), depending only on \(R\), with the following property. Let \(J\ge 0\) bound the interaction terms, let \(\Delta E{\gt}0\) be a gap, let \(\zeta \) be the decay length of Lemma 1 obtained by raising the Lieb–Robinson velocity by a factor \(\tfrac 54\), and put \(\xi '=6\zeta \) and \(C_1=\max (K\zeta ^3,1)\). For every chain of \(L\) sites with local dimension \(D\ge 1\) and every gapped ground vector \(\Omega \) of a Hamiltonian of range \(R\), strength \(J\) and gap \(\Delta E\), with \(k_0=\lceil \exp (2S_{\max })/2\rceil \) for the bound \(S_{\max }\) of the area law at the same \(\xi '\) and \(C_1\), as in Theorem 32.22.5, truncating every cut onto the \(k'\) leading eigenvectors of its left marginal, for every natural number \(k'\ge k_0\), gives an open-boundary matrix product state \(\Phi \) of bond dimension at most \(k'\) with
For \(D=1\) the exponent is read as \(0\); the bound then holds because \(\Phi =\Omega \), as in Theorem 32.14.6. The constants depend only on \(R\), \(J/\Delta E\) and \(D\), not on \(L\). This is the conclusion of the section on matrix product states of [ 240 ] under its Hamiltonian assumptions alone, at the printed rate \(\xi '=6\xi \) for \(\xi =\zeta \) and with \(C_1\) polynomial in \(\zeta \). The length \(\zeta \) may stand in for the source’s \(\xi \) because the source fixes \(\xi =\max (2v/\Delta E,\xi _C)\) only through some Lieb–Robinson pair \((v,\xi _C)\) (lines 96–112), and raising \(v\) keeps a pair valid [ 137 ] . The chain estimate behind \(\zeta \), whose prefactor grows with \(J\) and whose second operator acts on one site, is not shown to give a Lieb–Robinson pair in the source’s form [ 137 , 148 ] . The error is measured in the squared norm [ 146 ] , and the constant is that of Theorem 32.14.4 [ 132 ] .
Let \(R\) be a range and \(\kappa {\gt}6\). There is a constant \(K\ge 0\), depending only on \(R\) and \(\kappa \), with the following property. Let \(J\ge 0\) bound the interaction terms, let \(\Delta E{\gt}0\) be a gap, let \(\hat\xi \) be the decay length of Lemma 1 in units of the gap, and put \(\xi '=\kappa \hat\xi \) and \(C_1=\max (K\hat\xi ^3,1)\). For every chain of \(L\) sites with local dimension \(D\ge 1\) and every gapped ground vector \(\Omega \) of a Hamiltonian of range \(R\), strength \(J\) and gap \(\Delta E\), with \(k_0=\lceil \exp (2S_{\max })/2\rceil \) for the bound \(S_{\max }\) of the area law, truncating every cut onto the \(k'\) leading eigenvectors of its left marginal, for every natural number \(k'\ge k_0\), gives an open-boundary matrix product state \(\Phi \) of bond dimension at most \(k'\) with
For \(D=1\) the exponent is read as \(0\); the bound then holds because \(\Phi =\Omega \), as in Theorem 32.14.6. The constants depend only on \(R\), \(\kappa \), \(J/\Delta E\) and \(D\), not on \(L\), and \(C_1\) is polynomial in \(\hat\xi \). This is the form at the length \(\hat\xi \), at any rate \(\xi '=\kappa \hat\xi \) with \(\kappa {\gt}6\); the printed rate is Theorem 32.14.9. The length \(\hat\xi \) is not shown to be the length of the source’s Lieb–Robinson pair [ 137 , 148 ] . The error is measured in the squared norm [ 146 ] , and the constant is that of Theorem 32.14.4 [ 132 ] .
32.15 Correlations of Schmidt-diagonal operators in Hastings’ area law
Lemma 3 of [ 240 ] bounds the correlation between an operator that is diagonal in the Schmidt basis across a cut and an operator supported away from a buffer around the cut. Its proof uses only the approximate factorization of the ground-state projector supplied by Lemma 1. The two statements below take that factorization as a hypothesis, with an error \(\varepsilon \) that is left free.
Let \(\Omega \) be a unit vector and \(P=|\Omega \rangle \langle \Omega |\). Let \(O_B,O_L,O_R\) be positive semidefinite operators of norm at most one with \(\| O_BO_LO_R-P\| \le \varepsilon \), and let \(A,B_L,B_R\) be operators of norm at most one. Suppose that \(O_B\) commutes with \(A\), \(O_R\) with \(B_L\), \(B_R\) with \(O_L\) and \(O_L\) with \(O_R\), and that \(B_L\Omega =B_R\Omega \). Then
Since \(\| O_BO_LO_R\Omega -\Omega \| \le \varepsilon \) and the factors are positive contractions, \(\| (O_B-1)\Omega \| \), \(\| (O_L-1)\Omega \| \) and \(\| (O_R-1)\Omega \| \) are at most \(\delta =\sqrt{2\varepsilon }\). Writing \(x\approx _\eta y\) for \(|x-y|\le \eta \),
The first three steps use \(\| (O_R-1)\Omega \| \), \(\| (O_B-1)\Omega \| \) and \(\| (O_L-1)\Omega \| \) with norms at most one, the two equalities use the commutations and \(B_L\Omega =B_R\Omega \), and the last uses \(\| O_BO_LO_R-P\| \le \varepsilon \). Finally \(\langle \Omega ,APB_R\Omega \rangle =\langle \Omega ,A\Omega \rangle \langle \Omega ,B_R\Omega \rangle =\langle \Omega ,A\Omega \rangle \langle \Omega ,B_L\Omega \rangle \), and the total cost is \(3\sqrt{2\varepsilon }+\varepsilon \).
Let \((\psi (\alpha ))_\alpha \) be an orthonormal family and \(|O(\alpha )|\le 1\). Then \(\sum _\alpha O(\alpha )\, |\psi (\alpha )\rangle \langle \psi (\alpha )|\) has norm at most one.
By Bessel’s inequality, \(\| \sum _\alpha O(\alpha )\langle \psi (\alpha ),v\rangle \psi (\alpha )\| ^2 =\sum _\alpha |O(\alpha )|^2|\langle \psi (\alpha ),v\rangle |^2\le \| v\| ^2\).
Let \(\Omega \) be a unit vector on a chain of \(L\) sites, with Schmidt decomposition \(\Omega =\sum _\alpha a_\alpha \, \psi _L(\alpha )\otimes \psi _R(\alpha )\) across the cut after \(j\) sites. Let \(B_L=\sum _\alpha O(\alpha )\, |\psi _L(\alpha )\rangle \langle \psi _L(\alpha )| \otimes 1\) with \(|O(\alpha )|\le 1\), and let \(A\) be an operator of norm at most one supported away from the buffer, the sites among \(j-l,\dots ,j+l-1\) that lie in the chain (fewer than \(2l\) of them when the cut is within \(l\) sites of an end of the chain). Suppose there are positive semidefinite operators \(O_B,O_L,O_R\) of norm at most one, supported on the buffer, on the first \(j\) sites and on the remaining sites respectively, with \(\| O_BO_LO_R-|\Omega \rangle \langle \Omega |\| \le \varepsilon \). Then
This is a stage of the argument: the factorization is a hypothesis. With the factorization of Lemma 1 for a gapped chain it is Lemma 3 of [ 240 ] , equation (fwdback), with the error of that factorization in place of \(\varepsilon \).
The operator \(B_R=1\otimes \sum _\alpha O(\alpha )\, |\psi _R(\alpha )\rangle \langle \psi _R(\alpha )|\) satisfies
by the Schmidt decomposition and orthonormality, and both \(B_L\) and \(B_R\) have norm at most one by Theorem 32.15.2. Operators supported on disjoint sets of sites commute, so the supports give the four commutations of Theorem 32.15.1, which then applies.
Let \(R\) be a range. There is a constant \(K\ge 0\), depending only on \(R\), with the following property. Let \(H\) be a Hamiltonian on a chain of \(L\) sites, a sum of terms of range \(R\) and norm at most \(J\ge 0\), whose ground state \(\Psi _0\) is unique with gap \(\Delta E{\gt}0\) above the ground energy, let \(\tilde\xi \) be the length of Theorem 32.4.14, and let \(\Psi _0=\sum _\alpha a_\alpha \, \psi _L(\alpha )\otimes \psi _R(\alpha )\) with orthonormal \(\psi _L(\alpha )\) on the first \(j\) sites and orthonormal \(\psi _R(\alpha )\) on the rest. Let \(B_L=\sum _\alpha O(\alpha )\, |\psi _L(\alpha )\rangle \langle \psi _L(\alpha )| \otimes 1\) with \(|O(\alpha )|\le 1\), and let \(A\) be an operator of norm at most one supported away from the sites \(j-l,\dots ,j+l-1\). Then
This is a quantitatively restricted form of [ 240 , Lemma 3 ] , equation (fwdback), lines 627–646, with Lemma 1 at the printed rate \(\xi '=6\tilde\xi \) and the constant \(C_1=K\tilde\xi ^3\) bounded by a polynomial in the length. The length \(\tilde\xi \) may stand in for the source’s \(\xi \): the source fixes \(\xi =\max (2v/\Delta E,\xi _C)\) only through some Lieb–Robinson pair, and a bound valid for \(|t|\le l/v\) stays valid for a larger velocity [ 240 , lines 96–112 ] , so raising the velocity of the chain estimate is a legitimate choice [ 137 ] . The proof needs \(O_B\) positive, and the positive form of Lemma 1 decays at half the rate, so \(\epsilon (l)\) decays at the rate \(12\tilde\xi \) [ 175 ] . It also inherits the remaining restriction of the formalized Lemma 1: the length \(\tilde\xi \) comes from the chain Lieb–Robinson estimate with raised velocity, and is not shown to be the source’s \(\max (2v/\Delta E,\xi _C)\) for a Lieb–Robinson pair in the source’s sense [ 137 , 148 ] . The bound is on the modulus of the difference, which is stronger than the printed one-sided bound.
The positive form in Theorem 32.4.14 gives, at the cut \(j\) and length \(l\), a positive semidefinite \(O_B\) of norm at most one on the sites \(j-l,\dots ,j+l-1\) and commuting orthogonal projections \(O_L\), \(O_R\) on the two sides of the cut with \(\| O_BO_LO_R-P_0\| \le \epsilon (l)\). Orthogonal projections are positive semidefinite contractions, so Theorem 32.15.3 applies with \(\varepsilon =\epsilon (l)\).
32.16 The positive version of Hastings’ approximate factorization
Lemma 1 of [ 240 ] asserts positive operators, but its construction gives an operator \(O_B\) that need not be positive. The note added to the source replaces \(O_B\) by \(O_B^\dagger O_B\). This section records that step on the chain of \(L\) sites, with \(P_0=|\Omega \rangle \langle \Omega |\) for a unit vector \(\Omega \) and operator norms throughout.
Let \(c\) and \(l\) be natural numbers and \(\epsilon \) real. Operators \(O_B,O_L,O_R\) form an approximate factorization at \((c,l)\) with error \(\epsilon \) if \(O_L\) and \(O_R\) are commuting orthogonal projections supported on the first \(c\) sites and on the remaining sites, \(O_B\) has norm at most one and is supported on the sites \(c-l,\dots ,c+l-1\), and
An approximate factorization at \((c,l)\) with error \(\epsilon \) is one with every error \(\epsilon '\ge \epsilon \).
Only the bound 45 involves the error.
If an operator is supported on a set of sites, so is its adjoint.
The matrix entries of the adjoint are the conjugates of the transposed entries, and the support conditions are symmetric in the two configurations.
For a unit vector \(\Omega \), \(|\Omega \rangle \langle \Omega |\) is an orthogonal projection.
It is self-adjoint, and its square is \(\langle \Omega ,\Omega \rangle \, |\Omega \rangle \langle \Omega |=|\Omega \rangle \langle \Omega |\).
For \(0\le \epsilon \le 1\),
\(1-(1-\epsilon )^2\le 2\epsilon \), and \(\epsilon ^2\le \epsilon \le \sqrt\epsilon \).
If \(O_B,O_L,O_R\) form an approximate factorization at \((c,l)\) with error \(\epsilon \), then \(O_B^\dagger O_B\) is positive semidefinite and \(O_B^\dagger O_B,O_L,O_R\) form an approximate factorization at \((c,l)\) with error \((\sqrt2+4)\sqrt\epsilon \).
The operator \(O_B^\dagger O_B\) is positive semidefinite, has norm \(\Vert O_B\Vert ^2\le 1\), and is supported where \(O_B\) is. The positivity correction of the note added, with \(Q=O_LO_R\), bounds the error by \(\sqrt{1-(1-\epsilon )^2}+3\epsilon +\epsilon ^2\), which is at most \((\sqrt2+4)\sqrt\epsilon \) by 46 when \(\epsilon \le 1\). When \(\epsilon {\gt}1\), the error is at most \(\Vert O_B^\dagger O_BO_LO_R\Vert +\Vert P_0\Vert \le 2{\lt}(\sqrt2+4)\sqrt\epsilon \).
Let \(\Omega \) be a unit vector, \(C_1\ge 0\) and \(\xi '\) real. Suppose that at a cut \(c\) and length \(l\) there is an approximate factorization with error \(C_1e^{-l/\xi '}\). Then at the same \((c,l)\) there is one with \(O_B\) positive semidefinite and error
This is the positive operator \(O_B^+\) of the note added to [ 240 ] , lines 912–943, with the rate of Lemma 1 halved [ 175 ] .
Apply Theorem 32.16.6 and use \(\sqrt{C_1e^{-l/\xi '}}=\sqrt{C_1}\, e^{-l/(2\xi ')}\).
The sites \(j-m,\dots ,j+m-1\) lie among the first \(j+m\) sites.
32.17 Spectral cut-offs in Hastings’ approximate ground-state projector
In the proof of Lemma 1 of [ 240 ] , the localized filtered terms \(M_L\) and \(M_R\) nearly annihilate the ground vector, and the operators \(O_L\) and \(O_R\) are their spectral projections onto the small eigenvalues. This section proves that such a cut-off moves a vector by at most the norm of the vector’s image under \(M\) divided by the threshold, that it reduces \(M\) to norm at most the threshold, and that the cut-offs of operators supported on disjoint regions are supported there and commute. The source cuts off at the eigenvalues at most the threshold; here the cut-off keeps the eigenvalues of absolute value at most the threshold, because the terms need not be positive and the source later uses the norm bound on \(M_LO_L\) [ 184 ] . Norms of operators are operator norms.
For a Hermitian matrix \(M\) and a real number \(\theta \), the spectral cut-off \(O_\theta (M)\) is the spectral projection of \(M\) onto its eigenvalues of absolute value at most \(\theta \), that is \(f(M)\) for the function \(f\) equal to \(1\) on \([-\theta ,\theta ]\) and to \(0\) elsewhere.
For a Hermitian matrix \(M\) and every \(\theta \), the cut-off \(O_\theta (M)\) is an orthogonal projection.
The function \(f\) is real and satisfies \(f^2=f\).
Let \(M\) be Hermitian and \(\theta {\gt}0\). For every vector \(\Psi \),
The threshold \(\theta \) is free. The source bounds \(\lVert M_L\Psi _0\rVert \) by \(C\Delta E^{-1}J^2e^{-l/3\xi }\) for an unspecified constant \(C\) of order one. With \(\theta =\Delta E^{-1}J^2e^{-l/6\xi }\) this gives \(\lVert (O_L-1)\Psi _0\rVert \le Ce^{-l/6\xi }\), the bound \(O(e^{-l/6\xi })\) of the source, and likewise for \(O_R\); the constant, the residual and the threshold are fixed where Lemma 1 is assembled.
Let \(h(t)=1/t\) for \(\lvert t\rvert {\gt}\theta \) and \(h(t)=0\) otherwise. Then \(1-O_\theta (M)=h(M)M\), since \(t\, h(t)=1-f(t)\) for every real \(t\). The eigenvalues of \(h(M)\) lie in \([-1/\theta ,1/\theta ]\), so \(\lVert h(M)\rVert \le 1/\theta \) and \(\lVert \Psi -O_\theta (M)\Psi \rVert =\lVert h(M)M\Psi \rVert \le \lVert M\Psi \rVert /\theta \).
Let \(M\) be Hermitian and \(\theta \ge 0\). Then \(\lVert M\, O_\theta (M)\rVert \le \theta \).
The operator \(M\, O_\theta (M)\) is \(g(M)\) for \(g(t)=t\, f(t)\), and \(\lvert g(t)\rvert \le \theta \) for every real \(t\).
Let \(O\) and \(O'\) be commuting orthogonal projections with \(\lVert AO\rVert \le a\) and \(\lVert BO'\rVert \le b\). Then \(\lVert (A+B)OO'\rVert \le a+b\). In particular, if \(M_L\) and \(M_R\) are Hermitian, \(\theta \ge 0\), and their cut-offs \(O_L\) and \(O_R\) at \(\theta \) commute, then \(\lVert (M_L+M_R)O_LO_R\rVert \le 2\theta \).
Since \(OO'=O'O\), \((A+B)OO'=(AO)O'+(BO')O\), and projections have norm at most one.
If \(O\) is an orthogonal projection and \(O'\) any operator, then for every vector \(\Psi \), \(\lVert \Psi -OO'\Psi \rVert \le \lVert \Psi -O\Psi \rVert +\lVert \Psi -O'\Psi \rVert \).
Write \(\Psi -OO'\Psi =(\Psi -O\Psi )+O(\Psi -O'\Psi )\) and use \(\lVert O\rVert \le 1\).
Let \(K\) be a Hermitian operator on the sites of a region \(D\), and let \(K\otimes 1\) be its extension by the identity outside \(D\). Then \(O_\theta (K\otimes 1)=O_\theta (K)\otimes 1\). In particular, the cut-off of an operator supported on \(D\) is supported on \(D\).
The map \(K\mapsto K\otimes 1\) is a unital homomorphism of star algebras, and such homomorphisms commute with the continuous functional calculus. The spectrum is finite, so \(f\) is continuous on it.
Let \(K\) and \(K'\) be Hermitian operators on disjoint regions \(X\) and \(Y\). For all thresholds \(\theta \) and \(\theta '\), the cut-offs \(O_\theta (K\otimes 1)\) and \(O_{\theta '}(K'\otimes 1)\) commute. Applied to \(M_L\), supported left of the cut, and \(M_R\), supported right of it, this shows that \(O_L\) and \(O_R\) are commuting orthogonal projections.
By Theorem 32.17.7 the two cut-offs are extensions by the identity of operators on \(X\) and on \(Y\), and such extensions commute when \(X\) and \(Y\) are disjoint. A Hermitian operator supported on a region is such an extension of a Hermitian operator on that region, so the same holds for Hermitian operators supported on \(X\) and on \(Y\).
32.18 The decay length of Hastings’ approximate factorization
The final estimate of the proof of Lemma 1 of [ 240 ] , lines 903–907, bounds the error of the approximate factorization by a multiple of \(\sqrt{l/\xi }\, e^{-l/6\xi }\). The prefactor \(\sqrt l\) cannot be absorbed into a constant at the rate \(e^{-l/6\xi }\), but it can at every slower rate \(e^{-l/\xi '}\) with \(\xi '{\gt}6\xi \). At a fixed length this is all the argument gives. The printed rate \(\xi '=6\xi \) is reached by choosing the length anew, with the velocity of the chain estimate raised by the factor \(\tfrac 54\) (Theorem 32.4.14); the chain estimate is not shown to give a Lieb–Robinson pair in the source’s sense, before or after the raise.
For \(\delta {\gt}0\) and \(l\ge 0\),
With \(x=2\delta l\), the inequality \(x\le e^{x-1}\) gives \(xe^{-x}\le e^{-1}\), that is \(2e\delta \, l\, e^{-2\delta l}\le 1\), which is the square of 49. The bound is attained at \(l=1/(2\delta )\).
For \(\delta {\gt}0\), a natural number \(p\), a real number \(r\) and \(l\ge 0\),
With \(x=\delta l/p\) for \(p\ge 1\), \(l^pe^{-\delta l}=(p/\delta )^p(xe^{-x})^p\) and \(xe^{-x}\le e^{-1}\). The second bound follows from the first after splitting off \(e^{-rl}\).
Let \(\xi {\gt}0\), \(\xi '{\gt}6\xi \) and \(\delta =1/(6\xi )-1/\xi '\). For \(a,b\ge 0\) and \(l\ge 0\),
Applied to the final estimate of the proof of Lemma 1, this gives the approximate factorization with error \(C_1e^{-l/\xi '}\) for every \(\xi '{\gt}6\xi \), with \(C_1\) depending on \(\xi '-6\xi \), rather than at the printed \(\xi '=6\xi \) of [ 240 , lines 141–142 ] .
Write \(e^{-l/6\xi }=e^{-l/\xi '}e^{-\delta l}\). Then \(a\, e^{-\delta l}\le a\), and \(b\sqrt l\, e^{-\delta l}\le b/\sqrt{2e\delta }\) by 49.
Let \(\xi {\gt}0\), \(\xi '{\gt}6\xi \), \(\delta =1/(6\xi )-1/\xi '\) and \(a,b\ge 0\). Suppose that at a cut \(c\) and length \(l\) there is an approximate factorization with error \((a+b\sqrt l)\, e^{-l/6\xi }\). Then at the same \((c,l)\) there is one with error
This is a stage of Lemma 1 of [ 240 ] : the factorizations with the square-root error, the form the proof of Lemma 1 reaches at lines 903–907, are a hypothesis, and the theorem only converts their error to the form \(C_1e^{-l/\xi '}\) of the conclusion of Lemma 1, for a decay length \(\xi '{\gt}6\xi \) in place of the printed \(\xi '=6\xi \) [ 148 ] .
Under the hypotheses of Theorem 32.18.4, with \(\Omega \) a unit vector and \(C_1=a+b/\sqrt{2e\delta }\), there is at the same \((c,l)\) an approximate factorization with \(O_B\) positive semidefinite and error \((\sqrt2+4)\sqrt{C_1}\, e^{-l/(2\xi ')}\).
32.19 The boundary operator of Hastings’ approximate factorization
The last two steps of the proof of Lemma 1 of [ 240 ] turn the Gaussian time average of the Hamiltonian, which is close to the ground-state projector on the range of the cut-offs \(O_L\) and \(O_R\), into an operator \(O_B\) supported near the cut. Write \(P_\eta (H)=\int e^{iHt}\, d\gamma _\eta (t)\) for the Gaussian time average of variance \(\eta \), and let \(a\) and \(v\) be the constants of the chain Lieb–Robinson bound for terms of range \(R\) and norm at most \(J\).
Let \(H\) be a chain Hamiltonian of range \(R\) and term norm \(J\) on \(L\) sites with local dimension \(q\ge 1\), let \(E_0\) be real and \(H'=H-E_0\), let \(H_0\) be Hermitian with \(H'-H_0\) supported on a set \(Z\) of sites, let \(O_L\) and \(O_R\) be orthogonal projections, and let \(P\) be an operator on the chain. Suppose
Then for every \(\ell '\ge 0\) there is an operator \(O_B\) supported on the sites within distance \(\ell '\) of \(Z\), with \(\| O_B\| \le 1\) and
With \(E_0\) the ground energy, which the source sets to zero (line 767), \(H_0=M_L+M_R\) and the cut-offs of the localized parts, this is the operator \(O_B\) of Lemma 1 of [ 240 ] , constructed at lines 881–901.
Put \(A=H_0\) and \(B=H'-H_0\), so that \(A+B=H'\), and let \(P_B=\int e^{iH't}e^{-iH_0t}\, d\gamma _\eta (t)\) be the Gaussian average of the interaction-picture propagator. Shifting \(H_0\) by \(E_0\) changes neither \(P_B\), since the two phases cancel, nor \(H'-H_0\), and with the shifted operator \(A+B=H\). Replacing \(P_\eta (H')\) by \(P_B\) in front of \(O_LO_R\) costs at most \(\sqrt\eta \, \| H_0O_LO_R\| \), so \(\| P_BO_LO_R-P\| \le \varepsilon _0+\sqrt\eta \, \theta \). The Lieb–Robinson bound for \(H\) localizes the generator \(e^{iHs}(H'-H_0)e^{-iHs}\) of the propagator onto the \(\ell '\)-neighbourhood of \(Z\), which gives \(O_B\) with \(\| O_B\| \le 1\) and \(\| P_B-O_B\| \) at most the last two terms of (54). Since \(\| O_LO_R\| \le 1\),
32.20 The entropy gain across a cut in Hastings’ area law
The doubling step of [ 240 ] rests on a lower bound for the mutual information of the two intervals of \(l\) sites beside a cut. The approximate factorization of the ground-state projector of [ 240 , Lemma 1 ] yields an effect with a large expectation in the ground state and a small expectation in the product of the two marginals, and the monotonicity of the relative entropy turns this into the bound. The effect is \(E=O_B^\dagger O_B\), as in the note added to the source (route 1). The constants are explicit: \(C_1\) is replaced by \(2C_1\) and the constant \(C_2\) is \(7\), as recorded in [ 135 ] .
Let \(\Omega \) be a unit vector on the chain of \(L\) sites with local dimension \(q\), let \(c\) be a cut, let \(l\) be a natural number, and let \(\xi '{\gt}0\) and \(C_1\ge 1\). Write \(A\) for the \(l\) sites before the cut and \(B\) for the \(l\) sites after it, cut short at the ends of the chain when fewer than \(l\) sites remain there, \(P_0=|\Omega \rangle \langle \Omega |\), and \(\epsilon =C_1e^{-l/\xi '}\). Suppose there are commuting orthogonal projections \(O_L\) supported on the first \(c\) sites and \(O_R\) supported on the sites after them, and an operator \(O_B\) of norm at most one supported on \(A\cup B\), with
Let \(\sigma =\rho _{{\lt}c}\otimes \rho _{\ge c}\) be the product of the two marginals of \(P_0\) across the cut, and suppose \(\operatorname {tr}(\sigma P_0)\le 2\epsilon \). Then, with \(\epsilon '=2C_1e^{-l/\xi '}\),
This is a stage of eq. (relent) of [ 240 ] , lines 234–242, with the constants of [ 135 ] : the factorization of Lemma 1 and the bound on the overlap are hypotheses.
The mutual information \(S(A)+S(B)-S(A\cup B)\) is nonnegative by subadditivity. If \(\epsilon '{\gt}1/68\), the right side of (56) is negative and there is nothing to prove. Let \(\epsilon '\le 1/68\) and \(E=O_B^\dagger O_B\). Its expectation in \(\Omega \) is \(\| O_B\Omega \| ^2\ge (1-2\epsilon )^2\ge 1-2\epsilon '\). The product \(Z=O_LO_R\) is a projection with \(\| ZEZ-P_0\| \le 2\epsilon +\epsilon ^2\) and \(\operatorname {tr}(\sigma Z)\ge 1-2\epsilon ^2\), and the Cauchy–Schwarz inequality for the expectation in \(\sigma \) bounds \(x=\operatorname {tr}(\sigma E)\) by \(36\epsilon '\). Write \(\rho =P_0\) for the ground state, with marginals \(\rho _{A\cup B}\), \(\rho _A\) and \(\rho _B\). Since \(E\) is supported on \(A\cup B\), its expectations in \(\rho \) and \(\sigma \) are those in the marginal \(\rho _{A\cup B}\) and in \(\rho _A\otimes \rho _B\). The support of \(\rho _{A\cup B}\) lies in the tensor product of the supports of \(\rho _A\) and \(\rho _B\), so the positive expectation of \(E\) in \(\rho _{A\cup B}\) makes its expectation \(x\) in \(\rho _A\otimes \rho _B\) positive as well. The monotonicity of the relative entropy under the two-outcome measurement \(\{ E,1-E\} \) bounds \(S(A)+S(B)-S(A\cup B)\) below by the binary relative entropy of the two expectations, which is decreasing in \(x\) and increasing in the ground-state expectation on the relevant range. Evaluated at \(1-2\epsilon '\) and \(36\epsilon '\le 36/68\), its two terms are at least \((1-2\epsilon ')\ln (1/\epsilon ')-5\) and \(-1\), which gives (56).
Let \(\Omega \), \(c\), \(l\), \(\xi '\), \(C_1\), \(A\) and \(B\) be as in Theorem 32.20.1, with \(q\ge 2\) and \(c+l\le L\). Suppose that for every natural number \(m\) with \(c+m\le L\) there are operators \(O_B\), \(O_L\), \(O_R\) as in Theorem 32.20.1, with \(O_B\) supported on the \(m\) sites on either side of the cut, cut short at the left end of the chain when \(m{\gt}c\), and \(\| O_BO_LO_R-P_0\| \le C_1e^{-m/\xi '}\). Suppose further that the entropy \(S_c\) of the first \(c\) sites satisfies
which for \(q\ge 2\) reads \(l\le (S_c-F(\xi ',q))/\ln q-\xi '\ln C_1\). This is the window assumption of [ 240 ] , line 483, with \(l\) in place of \(l_0\ge l\), with \(S_c\ge S_{\mathrm{cut}}\) in place of \(S_{\mathrm{cut}}\), and with the local dimension \(D\) written \(q\). Both replacements weaken the assumption. Then (56) holds. This is a stage of eq. (relent) of [ 240 ] for one pair of intervals, proved at lines 427–503 under that window assumption, with the constants of [ 135 ] . The factorization of Lemma 1 at every scale is a hypothesis, and the Hamiltonian does not enter; the source equation remains to be derived from Lemma 1.
The product \(\sigma \) of the two marginals is a mixture of pure states of product vectors, each of Schmidt rank one across the cut, so \(P=\operatorname {tr}(\sigma P_0)=\sum _iw_i|\langle \psi _i,\Omega \rangle |^2\). If \(P{\gt}2C_1e^{-l/\xi '}\), then \(2C_1^2/P{\lt}C_1e^{l/\xi '}\), and Theorem 32.12.15 with \(k=1\) gives \(S_c\le \xi '\ln (2C_1^2/P)\ln q+F(\xi ',q){\lt}(l+\xi '\ln C_1)\ln q+F(\xi ',q)\), against the hypothesis. Hence \(P\le 2C_1e^{-l/\xi '}\), and Theorem 32.20.1 applies.
Let \(R\) be a natural number. There is a constant \(K\ge 0\), depending only on \(R\), with the following property. Let \(\Omega \) be a gapped ground vector, with gap \(\Delta E{\gt}0\), of a Hamiltonian on the chain of \(L\) sites with local dimension \(q\), interaction range \(R\) and term norms at most \(J\ge 0\). Let \(\tilde\xi \) be the length of Theorem 32.4.14, \(\xi '=6\tilde\xi \) and \(C_1=K\tilde\xi ^3\). Let \(c\) be a cut and \(l\) a natural number with \(c+l\le L\), and let \(A\) and \(B\) be the \(l\) sites before and after the cut, cut short at the left end of the chain when \(l{\gt}c\). If the entropy \(S_c\) of the first \(c\) sites satisfies
then, with \(\epsilon '=2C_1e^{-l/\xi '}\),
This is eq. (relent) of [ 240 ] , lines 234–242, under the window assumption of line 483, with the constants of [ 135 ] . The rate is the printed \(\xi '=6\tilde\xi \) of lines 141–142, and \(C_1=K\tilde\xi ^3\) is polynomial in \(\tilde\xi \), as at lines 258–262. The length \(\tilde\xi \) may stand in for the source’s \(\xi \), because the source fixes \(\xi =\max (2v/\Delta E,\xi _C)\) only through some Lieb–Robinson pair \((v,\xi _C)\) (lines 96–112), and raising \(v\) keeps a pair valid. It has the form of the source’s \(\max (2v/\Delta E,\xi _C)\), but it comes from a Lieb–Robinson estimate on the chain whose prefactor grows with \(J\) and whose second operator acts on one site, and it is not shown to arise from a pair satisfying the bound of lines 96–112. This is recorded in [ 137 , 148 ] .
Theorem 32.4.14 gives \(K_0\ge 0\) such that at every cut and every length \(m\) there are operators \(O_B\), \(O_L\), \(O_R\) as in Theorem 32.20.2 with \(\| O_BO_LO_R-P_0\| \le K_0\tilde\xi ^3e^{-m/(6\tilde\xi )}\). Take \(K=K_0+1\). Since \(\tilde\xi \ge 8\), the constant \(C_1=K\tilde\xi ^3\) is at least \(1\), and the same operators satisfy the bound with \(C_1\) in place of \(K_0\tilde\xi ^3\). For \(q\ge 2\) Theorem 32.20.2 gives the claim. For \(q=1\) the entropy \(S_c\) vanishes while \(F(\xi ',1)=1+\ln (\xi '/2+1){\gt}0\), so the window assumption cannot hold.
32.21 The matrix product form of Trotter products
For the density matrix at nonzero temperature, [ 239 , lines 1024–1079 ] introduces a matrix product form in which every site carries one index, every site carries an operator acting on it alone, and the coefficients are products of functions of finite range. Every Trotter product of a finite-range Hamiltonian has this form exactly. The error between a Trotter product and the Gibbs operator is not estimated here. The source works on an arbitrary lattice; here the sites form a finite domain of the square lattice with the walk distance. The term supports lie in balls of radius \(R\) about specified centres, as in the source. The original diameter-bounded statement follows as a corollary. The source’s count of the indices at a site is not established [ 69 ] .
Let \(\Lambda \) be a finite domain with local dimension \(q\), and let \(l\ge 0\). An operator \(\rho \) on \(\mathcal H_\Lambda \) has a matrix product form of range \(l\) if there are a finite set \(\mathcal A\), operators \(\rho _j(a)\) on the site \(j\) for every site \(j\) and every \(a\in \mathcal A\), and functions \(F_i\colon \mathcal A^\Lambda \to \mathbb C\) for every site \(i\), such that \(F_i(\alpha )\) depends only on the indices \(\alpha _j\) with \(d_\Lambda (i,j)\le l\) and
This is the right-hand side of the product-form identity in [ 239 , lines 1024–1039 ] . The index set is the same at every site; a smaller index set at one site is padded with values on which the function at that site vanishes. A product form of range \(l\) has every larger range.
Fix \(i_0\in \Lambda \) and \(X\subseteq \Lambda \) such that \(d_\Lambda (i_0,j)\le R\) for every \(j\in X\). Every operator supported on \(X\) has a matrix product form of range \(R\). The centre need not belong to \(X\), and \(X\) may be empty. This is the finite planar-domain case of the supported-factor expansion in [ 239 , lines 1053–1061 ] .
Take \(\mathcal A=\{ 1,\dots ,q\} ^2\) and let the single-site operators be the matrix units \(|a_j\rangle \langle b_j|\). Attach to \(i_0\) the matrix element between the configurations equal to \(a\) and \(b\) on \(X\) and equal to \(a_{i_0}\) off \(X\). At every site outside \(X\), multiply the coefficient by the indicator of \(a_j=b_j\), including at \(i_0\) if \(i_0\notin X\). The matrix element reads only indices on \(X\) and at \(i_0\), all within radius \(R\); each diagonal indicator reads only its own index. Supported operators vanish between configurations differing off \(X\), and their other entries depend only on the labels on \(X\). Thus (60) holds entry by entry.
Let \(X\subseteq \Lambda \) be nonempty with \(d_\Lambda (x,y)\le R\) for all \(x,y\in X\). Every operator supported on \(X\) has a matrix product form of range \(R\).
Choose any \(i_0\in X\) and apply the preceding theorem.
If \(\rho \) and \(\sigma \) have matrix product forms of range \(l\), then so does \(\rho \sigma \). The identity has a matrix product form of every range, and hence so does every finite ordered product and every power of operators with matrix product forms of range \(l\).
With index sets \(\mathcal A\) and \(\mathcal B\), group the two indices at each site into one index in \(\mathcal A\times \mathcal B\), take the products \(\rho _j(a)\sigma _j(b)\) as the operators on the site \(j\), and the products of the two functions at each site as the new functions, as in [ 239 , lines 1068–1077 ] . The operators on distinct sites multiply factor by factor.
For each \(i\in \Lambda \), let \(H_i\) be an operator supported on \(X_i\subseteq \{ j:d_\Lambda (i,j)\le R\} \). For any ordered list \(i_1,\ldots ,i_m\), real \(\beta \), and integer \(n\ge 1\),
has a matrix product form of range \(R\). Neither Hermiticity nor nonempty supports are required. With the separate convention that the zero-step product is the identity, the conclusion also holds for \(n=0\). This proves the radius-based construction of [ 239 , lines 1053–1077 ] on the stated planar domains.
The supported operators form a finite-dimensional closed algebra, so each exponential is supported on \(X_i\). Apply the supported-operator theorem with centre \(i\), then closure under finite products and powers.
Let \(H=\sum _Xh_X\) be a Hamiltonian on \(\Lambda \) whose nonempty term supports have diameter at most \(R\), let \(X_1,\dots ,X_m\) be any finite sequence of supports, let \(\beta \) be real and let \(n\ge 1\). The Trotter product
has a matrix product form of range \(R\). With the separate convention \(\rho _0=1\), the same conclusion holds at \(n=0\). This is the diameter-restricted specialization of the construction in [ 239 , lines 1048–1077 ] .
The operators supported on a set of sites form a finite-dimensional, hence closed, algebra, so each factor \(\exp (-\beta h_{X_k}/n)\) is supported on \(X_k\). Each factor therefore has a matrix product form of range \(R\), and so does the product and its \(n\)-th power.
32.22 Hastings’ area law from the approximate factorization
The contradiction argument of Section 32.2 takes the window facts and the strengthened doubling inequality as hypotheses. This section discharges all of them from the approximate factorization of the ground-state projector, the first lemma of [ 240 ] . The removal of sites and the bounds on the window entropy come from Section 32, and the doubling inequality comes from the entropy gain across a cut of Section 32.20, whose proof contains the bootstrap lemma.
The entropy gain is proved when the entropy of the cut is large compared with the remainder \(F(\xi ',q)\) of the bootstrap lemma (Definition 32.12.8). The source meets this condition at [ 240 , lines 483–486 ] : when it fails, the bound \(S_{\max }\) is small, and the constant \(c_0\) in \(S_{\max }\) is chosen large enough. Here the window is lengthened instead, which needs the contradiction argument for every window at least as long as the doubling window.
Theorem 32.2.4 holds for every window length \(l_0\ge \lceil \xi _0\rceil _+2^M\), with \(S_{\max }=3l_0\ln q\).
If \(i+l_0{\gt}L\), Theorem 32.2.3 bounds the entropy by \((L-i)_+\ln q\le l_0\ln q\le S_{\max }\). Otherwise, suppose that \(S(\lbrack 0,i)){\gt}S_{\max }\). The assumed site-removal inequality gives, for \(i\le k\le i+l_0\),
Thus the strengthened doubling inequality applies throughout this window. Theorem 32.1.4 gives \(l_0{\lt}\lceil \xi _0\rceil _+2^M\le l_0\), a contradiction.
Let \(\xi '{\gt}0\), \(C_1\ge 1\) and \(q\ge 2\). Let \(\lceil \xi _0\rceil _+2^M\) be the doubling window of Definition 32.2.1 for the constants \(2C_1\) and \(C_2=7\) of the entropy gain, and put
For \(\xi '{\gt}0\), \(C_1\ge 1\) and \(q\ge 2\), the bound \(S_{\max }\) of Definition 32.22.2 is at most the bound of Definition 32.2.1 for the constants \(2C_1\) and \(7\), plus \(2(\xi '\ln C_1\ln q+F(\xi ',q))+3\ln q\). The added term is the part that [ 240 , lines 483–486 ] absorbs into the constant \(c_0\).
The maximum of two nonnegative numbers is at most their sum, and \(3\ln q\, \lceil x\rceil _+\le 3x\ln q+3\ln q\) for \(x\ge 0\).
Let \(\Omega \) be a gapped ground vector of a finite-range Hamiltonian on a chain of \(L\) sites with local dimension \(q\). Let \(\xi '{\gt}0\) and \(C_1\ge 1\), and suppose that for every cut \(c\) and every \(l\) with \(c+l\le L\) there are operators \(O_B\), \(O_L\), \(O_R\) with these properties:
\(O_L\) and \(O_R\) are commuting orthogonal projections acting on the first \(c\) sites and on the remaining sites;
\(\| O_B\| \le 1\), and \(O_B\) acts on the sites of \(\lbrack c-l,c+l)\) that lie in the chain;
\(\| O_BO_LO_R-|\Omega \rangle \langle \Omega |\| \le C_1e^{-l/\xi '}\).
Then the entropy of the first \(i\) sites is at most \(S_{\max }\) for every \(i\), with \(S_{\max }\) as in Definition 32.22.2. Supplying the factorization gives an area law by the argument of [ 240 ] , equation (entbd), with the quantitative changes below. Requiring \(C_1\ge 1\) loses nothing, since enlarging \(C_1\) only weakens that lemma. The bound \(S_{\max }\) grows as \(4^{\xi '\ln q}\), not \(2^{\xi '\ln q}\), as recorded in [ 207 ] . The entropy gain uses the constants \(2C_1\) and \(7\) from [ 135 ] .
For \(q\le 1\) every entropy vanishes. For \(q\ge 2\), apply Theorem 32.22.1 with the window length \(l_0\) of Definition 32.22.2 and the window entropy. Theorem 32.1 gives the first item, and Theorem 32.4 the second. For the third, fix a window all of whose cuts have entropy at least \(2S_{\max }/3=2l_0\ln q\), and let \(2l\le l_0\). Take an interval \(\lbrack a,a+2l)\) of largest entropy in the window. The choice of \(l_0\) gives
so Theorem 32.20.2 applies at the cut after \(a+l\) sites, with the constants \(2C_1\) and \(7\). It bounds the entropy of \(\lbrack a,a+2l)\) by the entropies of \(\lbrack a,a+l)\) and \(\lbrack a+l,a+2l)\), each at most the window entropy at length \(l\), minus the gain, which is the doubling inequality.
Let \(R\) be a range. There is a constant \(K\ge 0\), depending only on \(R\), with the following property. Let \(J\ge 0\) bound the norms of the interaction terms, let \(\Delta E{\gt}0\), and let \(\tilde\xi \) be the length of Theorem 32.4.14, which depends on \(R\) and \(J/\Delta E\) only. For every chain of \(L\) sites with local dimension \(q\), every finite-range Hamiltonian on it of range \(R\) with terms of norm at most \(J\), and every ground vector \(\Omega \) separated by a gap \(\Delta E\) from the rest of the spectrum, the entropy of the first \(i\) sites is at most \(S_{\max }\) for every \(i\), with \(S_{\max }\) as in Definition 32.22.2 for \(\xi '=6\tilde\xi \) and \(C_1=\max \{ K\tilde\xi ^3,1\} \). In particular \(S_{\max }\) depends on \(R\), \(J/\Delta E\) and \(q\), but not on \(L\), on the Hamiltonian, or on \(\Omega \).
This is the main theorem of [ 240 ] , equation (entbd), at the printed rate \(\xi '=6\xi \) for \(\xi =\tilde\xi \). The length \(\tilde\xi \) may stand in for the source’s \(\xi \) because the source fixes \(\xi =\max (2v/\Delta E,\xi _C)\) only through some Lieb–Robinson pair \((v,\xi _C)\) (lines 96–112), and raising \(v\) keeps a pair valid [ 137 ] . As in the source, the constant \(C_1\) is bounded by a polynomial in the length (lines 258–262). The source takes \(\xi =\max (2v/\Delta E,\xi _C)\) for a Lieb–Robinson pair \((v,\xi _C)\), which gives the printed value \(S_{\max }=c_0\xi '\ln (\xi ')\ln (D)2^{\xi '\ln D}\). Here, as in Theorem 32.4.14, \(\tilde\xi \) comes from a chain Lieb–Robinson estimate whose prefactor grows with \(J\) and whose second operator acts on a single site, with the velocity raised by the factor \(5/4\), so it is not shown to be the length of a Lieb–Robinson pair in the source’s sense (lines 96–112); see [ 137 , 148 ] . In addition \(S_{\max }\) grows as \(4^{\xi '\ln q}\), not \(2^{\xi '\ln q}\), as recorded in [ 207 ] , and the entropy gain enters with the constants of [ 135 ] . The printed value of \(S_{\max }\) is therefore not claimed.
Theorem 32.4.14 gives the constant \(K\), such that the operators of Theorem 32.22.4 exist at every cut and length with error \(K\tilde\xi ^3e^{-l/(6\tilde\xi )}\). Replace \(K\tilde\xi ^3\) by \(\max \{ K\tilde\xi ^3,1\} \), which only enlarges the error (Theorem 32.16.2), and apply Theorem 32.22.4 with \(\xi '=6\tilde\xi \).
32.23 Translation-invariant Hamiltonians on a finite set of sites
[ 239 ] shows that for a translation-invariant system on a lattice the low-energy commutators of the filtered terms are exponentially small in the linear size of the system. The statements here describe translation invariance on any finite set of sites \(V\), with a group \(G\) acting on \(V\), such as the translations of a periodic lattice or a group of automorphisms of a finite graph. Each \(g\in G\) moves the configurations \(x\colon V\to \{ 1,\dots ,q\} \) to \(g\cdot x=x\circ g^{-1}\), and \(P_g\) is the permutation matrix of this move.
For \(g\in G\), \(P_g\) is the permutation matrix of \(x\mapsto g\cdot x\) on the configurations. The map \(g\mapsto P_g\) is a unitary representation of \(G\), and \((P_gAP_g^\dagger )(x,y)=A(x\circ g,y\circ g)\) for every operator \(A\).
If \(A\) acts on the sites \(S\), then \(P_gAP_g^\dagger \) acts on \(g\cdot S\).
\(P_gAP_g^\dagger \) is \(A\) with both configurations precomposed with the permutation \(v\mapsto g\cdot v\) of the sites, which carries the support \(S\) to \(g\cdot S\).
An operator \(H\) is invariant under the action of \(G\) when \(P_gH=HP_g\) for every \(g\in G\). In particular, \(H\) is invariant if \(P_gHP_g^\dagger =H\) for every \(g\in G\).
Let \(G\) also act on a finite index set \(K\), and let the operators \(h_k\) satisfy \(h_{g\cdot k}=P_gh_kP_g^\dagger \). Then \(\sum _kh_k\) is invariant. In particular, for one operator \(H_0\), the sum \(\sum _{g\in G}P_gH_0P_g^\dagger \) of its translates is invariant.
Conjugating by \(P_g\) sends \(h_k\) to \(h_{g\cdot k}\), and \(k\mapsto g\cdot k\) permutes \(K\). For the translates, take \(K=G\) acting on itself by left multiplication.
For a unitary \(U\) and Hermitian \(H\), filtering \(UWU^\dagger \) with respect to \(UHU^\dagger \) gives \(U\widetilde WU^\dagger \). If \(H\) is invariant, the filtered operator of \(P_gWP_g^\dagger \) is therefore \(P_g\widetilde WP_g^\dagger \).
\(e^{iUHU^\dagger t}=Ue^{iHt}U^\dagger \), so the integrand of the filter is conjugated by \(U\) at every time, and conjugation by \(U\) commutes with the integral.
If every \(g\in G\) preserves the adjacency of a graph on \(V\), then \(d(g\cdot x,g\cdot y)=d(x,y)\) for all sites \(x\), \(y\). The rotations of a ring of \(n\) sites are such an action.
A walk from \(x\) to \(y\) is carried to a walk of the same length from \(g\cdot x\) to \(g\cdot y\), so \(d(g\cdot x,g\cdot y)\le d(x,y)\); apply this to \(g^{-1}\) for the reverse inequality.
32.24 Low-energy commutators under a supercell of translations
When the ground space is degenerate, the compressions \(PM_iP\) of the filtered terms to the ground space must nearly commute for the local approximations to determine a basis of ground states. For a translation-invariant system, [ 239 , lines 1603–1642 ] shows that their commutators are exponentially small in the linear size of the system. The source assumes that the momenta of the ground states are rational multiples of \(2\pi \) with denominators independent of the system size, and derives from this a supercell whose translations leave every ground state unchanged. Here such a subgroup of translations is assumed directly; the step from rational momenta to the supercell, and the geometry placing a translated site far away, are not formalized [ 182 ] .
Throughout, \(G\) acts on the finite set of sites, \(H\) is Hermitian and invariant, \(P\) is the orthogonal projection onto the eigenspace of \(H\) for an energy \(E_0\), and \(S\le G\) is a subgroup with \(P_gv=v\) for every \(g\in S\) and every \(v\) in this eigenspace.
Let \(Q\) be the orthogonal projection onto a closed subspace \(K\) of a Hilbert space, and let \(U^\dagger \) fix every vector of \(K\). Then \(Q(UWU^\dagger )Q=QWQ\) for every operator \(W\). In particular \(P(P_gWP_g^\dagger )P=PWP\) for every \(g\in S\).
\(U^\dagger Q=Q\), and taking adjoints \(QU=Q\); so \(QUWU^\dagger Q=QWQ\). For \(g\in S\) take \(U=P_g\): then \(U^\dagger =P_{g^{-1}}\) and \(g^{-1}\in S\).
Let \(G\) act on an index set, let the terms satisfy \(h_{g\cdot k}=P_gh_kP_g^\dagger \), and let \(M_k\) be their filtered operators. For every \(g\in S\) and all indices \(i\), \(j\),
Consequently, if \(B(i,j)\) bounds \(\| [PM_iP,PM_jP]\| \) for all \(i\) and \(j\), then \(B(i,g\cdot j)\) bounds it too, for every \(g\in S\).
Since \(H\) is invariant, \(M_{g\cdot j}=P_gM_jP_g^\dagger \), and its compression is \(PM_jP\) by the previous theorem.
With the bound of the low-energy commutator of separated operators on a finite graph, this gives the decay across supercell translations.
Let the sites be the vertices of a finite graph, let each term \(h_k\) act on a set \(X_k\), let the terms satisfy \(h_{g\cdot k}=P_gh_kP_g^\dagger \), and let every eigenvalue of \(H=\sum _kh_k\) equal \(E_0\) or be at least \(E_0+\Delta E\). If \(g\in S\) and \(X_{g\cdot j}\) is disjoint from the sites within distance \(l\) of \(X_i\), then \(\| [PM_iP,PM_jP]\| \) is at most the bound of Theorem 32.5.7 for the pair \(h_i\), \(h_{g\cdot j}\) at separation \(l\), which decays exponentially in \(l\).
The compression of \(M_k\) is that of \(h_k\), and the compression of \(h_{g\cdot j}=P_gh_jP_g^\dagger \) is that of \(h_j\) by Theorem 32.24.1. Apply the bound for separated operators to \(h_i\) and \(h_{g\cdot j}\).
Choosing \(g\in S\) that carries \(X_j\) to distance of order the linear size of the system from \(X_i\) gives the source’s conclusion. At line 1638 the source writes the first site as \((\vec x_1,y_2)\) where \((\vec x_1,y_1)\) is meant [ 182 ] .
32.25 A variational lower bound on the ground energy
For a translation-invariant Hamiltonian \(H=\sum _iH_i\) on \(N\) sites, whose terms \(H_i\) are translates of one another, [ 239 ] observes that the ground energy per site is at least the lowest eigenvalue of \(H_i\), and that the same holds after adding to \(H_i\) any combination of its iterated commutators with \(H\), because these commutators add up to zero over the translates. Write \(\operatorname {ad}_H(X)=HX-XH\), and \(\lambda _{\min }(K)\) for the lowest eigenvalue of a Hermitian \(K\), the largest \(\lambda \) with \(K-\lambda \ge 0\).
For Hermitian \(H\) and \(A\), a real number \(a\) and \(k\ge 0\), the operator \(a\, i^k\operatorname {ad}_H^k(A)\) is Hermitian.
By induction, \((\operatorname {ad}_H^kA)^\dagger =(-1)^k\operatorname {ad}_H^k(A)\), and \(\overline{i^k}(-1)^k=i^k\).
Let \(H_0\) be an operator, let \(U_1,\dots ,U_N\) be unitaries, and let \(H=\sum _iU_iH_0U_i^\dagger \) with every \(U_i\) commuting with \(H\). For real constants \(a_1,\dots ,a_m\) let
If \(K-\lambda \ge 0\), then \(H-N\lambda \ge 0\); in particular every eigenvalue of \(H\), and so its ground energy \(E_0\), is at least \(N\lambda \). This generalizes equation (varnum) of [ 239 ] , whose terms are the translates of one term; the source’s form is Theorem 32.25.3.
Conjugation by a unitary \(U\) that commutes with \(H\) commutes with \(\operatorname {ad}_H\), so \(\sum _iU_i\operatorname {ad}_H^k(H_0)U_i^\dagger =\operatorname {ad}_H^k(H)=0\) for \(k\ge 1\), and \(\sum _iU_iKU_i^\dagger =H\). Since \(U_i(K-\lambda )U_i^\dagger \ge 0\) and \(U_iU_i^\dagger =1\), summing gives \(H-N\lambda \ge 0\). For an eigenvector \(\Omega \) of \(H\) with eigenvalue \(E\), \(0\le \langle \Omega ,(H-N\lambda )\Omega \rangle =(E-N\lambda )\| \Omega \| ^2\).
Let a finite group \(G\) act on a finite set of sites, such as the translations of a periodic lattice of any dimension or a group of automorphisms of a finite graph, and let \(P_g\) be the unitary that moves the configurations of the sites along \(g\). For the sum \(H=\sum _{g\in G}P_gH_0P_g^\dagger \) of the translates of one term \(H_0\), and \(K\) as in Theorem 32.25.2, every eigenvalue of \(H\) is at least \(|G|\lambda \) whenever \(K-\lambda \ge 0\). For Hermitian \(H_0\), with \(\lambda =\lambda _{\min }(K)\) and \(N=|G|\),
Here \(N=|G|\) counts the summands. When \(G\) acts simply transitively on the sites, as the translations of a periodic lattice do, \(N\) is the number of sites. This is equation (varnum) of [ 239 ] , lines 856–877, for translation-invariant \(H=\sum _iH_i\); there the terms are written with \([H_i,H]=-\operatorname {ad}_H(H_i)\), which only changes the signs of the arbitrary constants.
The map \(g\mapsto P_g\) is multiplicative, so conjugation by \(P_h\) permutes the translates and fixes \(H\), and each \(P_h\) is a unitary commuting with \(H\). Theorem 32.25.2 applies. For Hermitian \(K\), its real spectrum is finite, and every spectral value is at least its infimum. The spectral theorem therefore gives \(K-\lambda _{\min }(K)\ge 0\), yielding the stated specialization.
The Gaussian expansion discussed at lines 1005–1009 of [ 239 ] corresponds, in the convention used here, to \(a_k=0\) for odd \(k\) and \(a_k=[(k-1)!!/k!](q/\Delta E^2)^{k/2}\) for even \(k\). The printed variance power requires correction, and \(i^k\) already supplies the alternating sign [ 64 ] . This infinite-series identity is separate from the finite-polynomial theorem above.
32.26 Counting connected clusters in a graph of bounded degree
The high-temperature cluster expansion of [ 239 ] sums over clusters of sites, and its convergence rests on the classical fact that a lattice of \(N\) sites carries at most \(Nx^{n}\) clusters of \(n\) sites for some constant \(x\) [ 239 , lines 1227–1254 ] . We prove this for an arbitrary finite simple graph whose vertices all have degree at most \(\Delta \), with the explicit constant \(x=\max (1,\Delta ^{2})\). A cluster is a set of vertices that induces a connected subgraph.
Let \(G\) be a finite simple graph in which every vertex has degree at most \(\Delta \), let \(v\) be a vertex, and let \(n\) be a natural number. At most \(\Delta ^{2(n-1)}\) sets of \(n\) vertices contain \(v\) and induce a connected subgraph of \(G\), where \(n-1\) is read as \(0\) when \(n=0\); no such set exists then.
Let \(S\) be such a set. Starting from \(\{ v\} \) and the walk of length zero, add the vertices of \(S\) one at a time: as long as the set \(T\) reached so far is a proper subset of \(S\), a path inside \(S\) from \(v\) to a vertex of \(S\setminus T\) leaves \(T\) along an edge \(wu\) with \(w\in T\) and \(u\in S\setminus T\). Inserting the detour \(w\to u\to w\) into the current closed walk at a visit of \(w\) adds two to its length and exactly the vertex \(u\) to its vertex set. After \(n-1\) steps this gives a closed walk from \(v\) of length \(2(n-1)\) whose vertex set is \(S\). The walk determines \(S\), so distinct sets give distinct walks, and at most \(\Delta ^{L}\) walks of length \(L\) start at \(v\), since each step has at most \(\Delta \) choices.
Let \(G\) be a simple graph on \(N\) vertices in which every vertex has degree at most \(\Delta \), and let \(n\) be a natural number. At most \(N\Delta ^{2(n-1)}\) sets of \(n\) vertices induce a connected subgraph of \(G\), with \(n-1\) read as \(0\) when \(n=0\). In particular the number is at most \(Nx^n\) for \(x=\max (1,\Delta ^2)\). The latter bound also holds for \(n=0\), since the empty induced graph is not connected.
A set inducing a connected subgraph is nonempty, so it contains some vertex; sum the bound of Theorem 32.26.1 over the \(N\) vertices. For \(n\ge 1\), \(\Delta ^{2(n-1)}=(\Delta ^2)^{n-1}\le x^{n-1}\le x^n\), since \(x\ge 1\). This choice also covers \(\Delta =0\), when the connected singleton sets would prevent taking \(x=0\).
32.27 The single-cluster bound in Hastings’ cluster expansion
Let \(H\) be a Hermitian Hamiltonian, let \(\beta \ge 0\), and put \(Z=\operatorname {tr}e^{-\beta H}\). For a cluster \(C\) and its disjoint boundary \(B(C)\), suppose that the Hermitian terms \(H_k\) have operator norm at most \(J\). As in the source, the terms are indexed by sites, with \(H_k\) supported near site \(k\), so \(C\) and \(B(C)\) are sets of sites. Write \(n_C=|C|\) and \(n_{B(C)}=|B(C)|\) for their numbers of sites, \(H_C=\sum _{k\in C}H_k\) and \(H_{B(C)}=\sum _{k\in B(C)}H_k\). Then for every operator \(O\)
This is the single-cluster estimate of [ 239 , Section IV ] , with an absolute value on the trace.
It is the finite-sum perturbation estimate in the companion quantum-channel volume [ 253 , “The single-cluster bound” ] , applied to \(T=C\cup B(C)\). The source proves it by an imaginary-time Dyson expansion. For the companion proof, put \(V=H_C+H_{B(C)}\) and \(a=(n_C+n_{B(C)})J\), and let \(I\) be the identity. The triangle inequality gives \(\| V\| \le a\). Hermiticity and monotonicity of the trace exponential then give
The matrix \(A=e^{-\beta (H-V)}\) satisfies \(A\ge 0\), so
where the second inequality is (68). For positive dimension, \(Z{\gt}0\), and dividing (69) by \(Z\) gives (66). In dimension zero the estimate uses \(0^{-1}=0\).
For a Hamiltonian \(H\) containing the terms \(H_k\), a cluster \(C\), its boundary \(B(C)\) and \(\beta \in \mathbb R\), put
By inclusion and exclusion this is the sum of the terms of the power series of \(e^{-\beta H}\) that contain every \(H_i\) with \(i\in C\) and no \(H_j\) with \(j\in B(C)\), the operator \(\rho (\beta ,C)\) of [ 239 , lines 1174–1177 ] . With \(E=C\setminus D\) it is \(\sum _{E\subseteq C}(-1)^{|C|-|E|}e^{-(Y+\sum _{i\in E}\beta H_i)}\) for \(Y=\beta (H-H_C-H_{B(C)})\).
Let \(H\) be Hermitian, let the terms \(H_k\) with \(k\in C\cup B(C)\) be Hermitian with \(\| H_k\| \le J\), and let \(\beta \ge 0\). Then, with \(Z=\operatorname {tr}e^{-\beta H}\),
The triangle inequality gives \(\| \beta (H_C+H_{B(C)})\| \le (n_C+n_{B(C)})J\beta \), and the bound for a Hermitian perturbation in the exponent in the companion quantum-channel volume [ 253 , “A Hermitian perturbation in the exponent” ] , applied to \(-\beta H+\beta (H_C+H_{B(C)})\), gives the claim.
Let \(\| H_k\| \le J\) for \(k\in C\cup B(C)\), let \(\beta \ge 0\), and suppose that \(n_C+n_{B(C)}\le Sn_C\). Then \(e^{(n_C+n_{B(C)})J\beta }\le \bigl(e^{SJ\beta }\bigr)^{n_C}\).
If \(C\cup B(C)\) is empty, both sides equal \(1\). Otherwise \(J\ge 0\), so \((n_C+n_{B(C)})J\beta \le Sn_CJ\beta \).
Let \(H\) be Hermitian, let the terms \(H_k\) with \(k\in C\cup B(C)\) be Hermitian with \(\| H_k\| \le J\), and let \(\beta \ge 0\). Then for every operator \(O\), with \(Z=\operatorname {tr}e^{-\beta H}\),
This is the bound stated before equation (percb2) in [ 239 , lines 1174–1195 ] , with \(\| O\| Z\) in place of the printed \(\operatorname {tr}(Oe^{-\beta H})\), which is the form used in (percb2), and with \(\| H_C+H_{B(C)}\| \le (n_C+n_{B(C)})J\). The printed form fails: applied to \(\pm O\) it would make \(\rho (\beta ,C)\) a multiple of \(e^{-\beta H}\) [ 179 ] .
By (70) with \(E=C\setminus D\), the alternating-sum bound of the companion quantum-channel volume [ 253 , “Alternating sums of shifted exponentials” ] , applied with \(Y=\beta (H-H_C-H_{B(C)})\) and the terms \(\beta H_i\), \(i\in C\), of norm at most \(\beta J\), gives
Theorem 32.27.3 gives \(\operatorname {tr}e^{-Y}\le e^{(n_C+n_{B(C)})J\beta }Z\).
Let \(C\) be a set of sites, and let \(B(C)\) be the set of \(j\notin C\) such that \(H_j\) acts on some site of \(C\). Suppose that every site \(i\in C\) has a set \(N(i)\) of at most \(S\) sites that contains \(i\) and every \(j\) for which \(H_j\) acts on \(i\). Then
In [ 239 , lines 181–183 ] , \(N(i)\) is the set of sites within the interaction range \(R\) of \(i\), of which there are at most \(S\), and \(H_j\) acts only on sites within distance \(R\) of \(j\). This is the second inequality of [ 239 , line 1186 ] . The same count holds with \(H_j\) acting on \(i\) replaced by any relation between terms. For the boundary of terms whose support meets that of \(H_C\), which the truncated expansion needs [ 129 ] , \(N(i)\) is the set of terms whose support meets that of \(H_i\). Their centres lie within \(2R\) of \(i\), so \(S\) counts the sites within distance \(2R\).
The sets \(C\) and \(B(C)\) are disjoint, and both lie in \(\bigcup _{i\in C}N(i)\), which has at most \(Sn_C\) elements.
In the setting of Theorem 32.27.5, suppose that \(n_C+n_{B(C)}\le Sn_C\). Then, with \(y=(e^{J\beta }-1)e^{SJ\beta }\),
This is equation (percb2) of [ 239 , lines 1186–1195 ] , multiplied by \(Z\). The source obtains \(n_C+n_{B(C)}\le Sn_C\) from the lattice, where \(S\) bounds the number of terms acting on a site; here it is a hypothesis, which Theorem 32.27.6 supplies.
Let \(C_1,\dots ,C_m\) be pairwise disjoint clusters with boundaries \(B(C_1),\dots ,B(C_m)\), each with \(n_{C_i}+n_{B(C_i)}\le Sn_{C_i}\), and let \(\rho (\beta ,C_1,\dots ,C_m)\) be the operator (70) for the cluster \(C_1\cup \dots \cup C_m\) and the boundary \(B(C_1)\cup \dots \cup B(C_m)\). Under the hypotheses of Theorem 32.27.5 on these terms,
This is the bound on \(\rho (\beta ,C_1,C_2,\dots )\) that the remark of [ 239 , lines 1218–1224 ] derives in the same way as (percb2). For the clusters of one term of the power series, no \(C_j\) meets \(B(C_i)\), so the operator above is the source’s \(\rho (\beta ,C_1,\dots ,C_m)\).
The union of the clusters has \(\sum _in_{C_i}\) sites, and the union of the boundaries has at most \(\sum _in_{B(C_i)}\) terms, so the union satisfies \(n_C+n_{B(C)}\le Sn_C\). Theorem 32.27.7 gives \(\| O\| y^{\sum _in_{C_i}}Z=\| O\| \prod _iy^{n_{C_i}}Z\).
In the setting of Theorem 32.27.7, let \(n\ge 1\) and \(0\le k\le n-1\), and put \(y=(e^{J\beta /n}-1)e^{SJ\beta /n}\). Then
This is the bound of [ 239 , lines 1322–1333 ] , multiplied by \(Z\), where \(\rho (\beta )=e^{-\beta H}\); the source proves it as (percb2).
Put \(Y=\beta (H-H_C-H_{B(C)})\). As in the proof of Theorem 32.27.5, \(\rho (\beta /n,C)\) is the alternating sum of the \(e^{-(Y+\sum _{i\in E}\beta H_i)/n}\) over \(E\subseteq C\). The bound for alternating sums between thermal factors in the companion quantum-channel volume [ 253 , “Alternating sums between thermal factors, uniform form” ] , with the weights \(k/n\), \(1/n\) and \((n-k-1)/n\) and the terms \(\beta H_i\) of norm at most \(\beta J\), gives
Theorem 32.27.3 gives \(\operatorname {tr}e^{-Y}\le e^{(n_C+n_{B(C)})J\beta }Z\). Taking \(1/n\)-th powers and using \(Z^{(n-1)/n}Z^{1/n}=Z\) bounds the right side by \(\| O\| (e^{J\beta /n}-1)^{n_C}e^{(n_C+n_{B(C)})J\beta /n}Z\), and Theorem 32.27.4, with \(\beta /n\) in place of \(\beta \), bounds \(e^{(n_C+n_{B(C)})J\beta /n}\) by \(e^{SJ\beta n_C/n}\).
32.28 The truncated expansion as an inclusion–exclusion sum
Expanding \(e^{-\beta H}=\sum _n\frac{(-\beta )^n}{n!}\bigl(\sum _kH_k\bigr)^n\) writes it as a sum over words \(H_{k_1}\cdots H_{k_n}\) in the terms of \(H\). In each word, terms whose supports meet are joined by active bonds, and the truncated operator \(\rho (\beta ,l_{\mathrm{proj}})\) keeps the words none of whose clusters has diameter larger than \(l_{\mathrm{proj}}\) [ 239 , lines 1090–1114 ] . The remark of [ 239 , lines 1218–1224 ] writes it as \(\rho -\rho _1+\rho _2-\cdots \), where \(\rho _m\) is the sum of \(\rho (\beta ,C_1,\dots ,C_m)\) over families of \(m\) large clusters. This section proves that identity for a finite set of terms. Which clusters count as large is an arbitrary property of sets of terms. The boundary of a cluster is the set of terms whose support meets the support of the cluster, which corrects the boundary of [ 239 , lines 1130–1133 ] ; with the printed boundary the identity fails [ 129 ] .
Let the terms be indexed by a finite set \(K\), and let \(G\) be a graph on \(K\). In the overlap graph two distinct terms are adjacent when their supports meet. The boundary \(\partial C\) of \(C\subseteq K\) is the set of terms outside \(C\) adjacent to a term of \(C\); in the overlap graph these are the terms outside \(C\) whose support meets the support of \(H_C\). A cluster of a set \(A\subseteq K\) of terms is a set \(C\subseteq A\) that is connected in \(G\) and such that \(A\) contains no term of \(\partial C\). A family \(U\) of sets of terms is admissible when no member meets the boundary of a member.
Two clusters of the same set that meet are equal. The members of an admissible family of connected sets are pairwise disjoint, the union of their boundaries is the boundary of their union, and if \(n_C+n_{\partial C}\le Sn_C\) for every member \(C\), then the union \(C_U=\bigcup _{C\in U}C\) has \(n_{C_U}+n_{\partial C_U}\le Sn_{C_U}\).
Let \(C\) be connected, meet \(C'\), and contain no term of \(\partial C'\). A path in \(C\) from a common term to any term of \(C\) cannot leave \(C'\), since its first step outside \(C'\) would land in \(\partial C'\). So \(C\subseteq C'\). For two clusters of \(A\) this applies in both directions. For two members of an admissible family it applies in both directions as well, which also covers a member strictly contained in another. The boundary of the union consists of the terms adjacent to the union and outside it, and admissibility puts every term of a member’s boundary outside the union. The count follows by adding the counts of the disjoint members.
In a ring that need not be commutative, \((\sum _{k\in s}a_k)^n\) is the sum of the ordered products \(a_{k_1}\cdots a_{k_n}\) over the words of length \(n\) in \(s\). For disjoint \(B,C\subseteq K\), the words of length \(n\) that contain every letter of \(C\) and no letter of \(B\) have ordered products summing to
The first statement is induction on \(n\), splitting off the first letter. For the second, for a word with letter set \(X\) disjoint from \(B\), \(\sum _{D\subseteq C,\ D\cap X=\emptyset }(-1)^{|D|}=\sum _{D\subseteq C\setminus X}(-1)^{|D|}\) is \(1\) when \(C\subseteq X\) and \(0\) otherwise.
Let \(H=\sum _{k\in K}H_k\) and let \(B\), \(C\) be disjoint. The words that contain every term of \(C\) and no term of \(B\), weighted by \((-\beta )^n/n!\), sum to the operator \(\rho (\beta ,C)\) of (70) with boundary \(B\).
By Theorem 32.28.3 the words of length \(n\) give \(\sum _{D\subseteq C}(-1)^{|D|}(H-H_B-H_D)^n\), and the exponential series of \(-\beta (H-H_B-H_D)\) sums each term.
Let \(T\) be the set of connected sets of terms that are large. The operator \(\rho (\beta ,l)\) is the sum over \(n\) of \((-\beta )^n/n!\) times the sum of the ordered products \(H_{k_1}\cdots H_{k_n}\) over the words of length \(n\) none of whose clusters is large. This is \(\rho (\beta ,l_{\mathrm{proj}})\) of [ 239 , lines 1101–1104 ] when large means diameter larger than \(l_{\mathrm{proj}}\).
For a set \(A\) of terms,
the sum over the admissible families \(U\subseteq T\) with \(\bigcup _{C\in U}C\subseteq A\) and \(A\cap \bigcup _{C\in U}\partial C=\emptyset \). These are exactly the families of large clusters of \(A\), so a set with \(k\) large clusters is counted by \(\binom km\) families of size \(m\).
A family of connected sets consists of clusters of \(A\) exactly when \(A\) contains its union and avoids the union of its boundaries, and such a family is admissible. The sum over the subsets of the set \(L\) of large clusters of \(A\) is \(1\) when \(L\) is empty and \(0\) otherwise.
Let \(H=\sum _{k\in K}H_k\). The series defining \(\rho (\beta ,l)\) converges, and
summed over the admissible families \(U\) of large connected sets, where \(\rho (\beta ,U)\) is the operator (70) for the union of \(U\) and the union of the boundaries. Grouping the families by size, this is \(\rho -\rho _1+\rho _2-\cdots \) of [ 239 , lines 1197–1225 ] , with the boundary of Definition 32.28.1 [ 129 ] .
Multiply the identity of Theorem 32.28.6 for the letter set of each word by its ordered product and exchange the finite sums. For each admissible \(U\), the words in which every member of \(U\) is a cluster are those containing the union and avoiding the union of the boundaries, which are disjoint, and Theorem 32.28.4 sums them. The empty family contributes \(e^{-\beta H}\).
Let \(H=\sum _kH_k\) be Hermitian with Hermitian terms of norm at most \(J\), let \(\beta \ge 0\), and let every large connected set \(C\) have \(n_C+n_{\partial C}\le Sn_C\). Then for every operator \(O\)
summed over the nonempty admissible families of large connected sets, with \(y=(e^{J\beta }-1)e^{SJ\beta }\) and \(Z=\operatorname {tr}e^{-\beta H}\).
32.29 The percolation sum in Hastings’ cluster expansion
At high temperature [ 239 ] compares the thermal operator \(\rho (\beta )\) with its truncation \(\rho (\beta ,l_{\rm proj})\), which keeps the terms of the power series of \(e^{-\beta H}\) whose clusters of active bonds all have diameter at most \(l_{\rm proj}\). The difference is an inclusion–exclusion sum over families of clusters of diameter larger than \(l_{\rm proj}\) [ 239 , lines 1196–1225 ] . A family of \(m\) such clusters \(C_1,\dots ,C_m\) contributes at most \(\prod _iy^{n_{C_i}}\) times \(\| O\| \), where \(n_C\) is the number of sites of \(C\) and \(y=(e^{J\beta }-1)e^{SJ\beta }\). This section bounds the resulting sum over families of clusters. The weights are kept abstract: a weight \(a_C\) with \(0\le a_C\le y^{n_C}\) stands for the contribution of the cluster \(C\), and a family \(U\) of clusters contributes \(\prod _{C\in U}a_C\). Every cluster is assumed to have at least \(l+1\) sites, for a size threshold \(l\).
The source starts the sum over cluster sizes at \(n=l_{\rm proj}+1\) [ 239 , line 1240 ] , which is right for nearest-neighbour bonds only. With terms of range \(R\), a cluster of \(n\) sites spans a distance of at most \(2Rn\), so a cluster of diameter larger than \(l_{\rm proj}\) has at least \(\lfloor l_{\rm proj}/(2R)\rfloor +1\) sites, and no more is true in general: on a chain the cluster \(\{ 0,2R\} \) has two sites and spans \(4R\). The size threshold for the application is therefore \(l=\lfloor l_{\rm proj}/(2R)\rfloor \) [ 42 ] .
Let \(T\) be a finite set with nonnegative weights \(a_C\), \(C\in T\), and let \(m\) be a natural number. Then
This is the bound on \(\rho _m\) in [ 239 , line 1210 ] .
Induction on \(T\). Write \(s=\sum _{C\in T}a_C\) and add an element \(b\notin T\) with weight \(a_b\). The \((m+1)\)-element subsets of \(T\cup \{ b\} \) are the \((m+1)\)-element subsets of \(T\) and the sets \(U\cup \{ b\} \) with \(U\) an \(m\)-element subset of \(T\). By the inductive hypothesis their sum is at most \(s^{m+1}/(m+1)!+a_bs^m/m!\), which is at most \((s+a_b)^{m+1}/(m+1)!\) since \(s^{m+1}+(m+1)a_bs^m\le (s+a_b)^{m+1}\) for \(s,a_b\ge 0\).
Let \(T\) be a finite set with nonnegative weights \(a_C\), \(C\in T\). Then
Grouped by the number \(m\) of clusters, the left side is the sum over \(m\ge 1\) of the left sides of Theorem 32.29.1, and the right side is \(\sum _{m\ge 1}(1/m!)(\sum _Ca_C)^m\), as in [ 239 , lines 1238–1241 ] .
Expanding the product gives the sum over all subsets, the empty one contributing \(1\). Then \(1+a_C\le e^{a_C}\) for each \(C\).
Let \(T\) be a finite family of clusters, where the cluster \(C\) has \(n_C\ge l+1\) sites, and suppose that at most \(Nx^k\) clusters of \(T\) have \(k\) sites, for every \(k\), where \(N\ge 0\). Let \(x,y\ge 0\) with \(xy{\lt}1\). Then
The right side is \(N\) times the geometric tail \(\sum _{k\ge l+1}(xy)^k\), the inner sum of [ 239 , lines 1238–1241 ] .
Group the clusters by their number of sites \(k\ge l+1\): the clusters with \(k\) sites contribute at most \(Nx^ky^k\). A finite sum of \((xy)^k\) over exponents \(k\ge l+1\) is at most \((xy)^{l+1}/(1-xy)\), the geometric series starting at \(k=l+1\).
In the setting of Theorem 32.29.3, let the weights satisfy \(0\le a_C\le y^{n_C}\), let \(N\ge 0\), and let \(0{\lt}xy\le 1/2\). Put \(\xi =1/\ln (1/(xy))\), a decay length in units of cluster size. Then
With the weights \(a_C\) bounding the contributions of the clusters, this is eq. (perc2) of [ 239 , lines 1227–1254 ] for the size threshold \(l\). With \(l=\lfloor l_{\rm proj}/(2R)\rfloor \), \(e^{-l/\xi }\le (xy)^{-1}e^{-l_{\rm proj}/\xi _{\rm perc}}\) with \(\xi _{\rm perc}=2R/\ln (1/(xy))\) in units of distance; this is the source’s decay length of order the interaction range [ 42 ] .
Let \(G\) be a simple graph on \(N\) vertices in which every vertex has degree at most \(\Delta \), and put \(x=\max (1,\Delta ^2)\). Let \(T\) be a finite family of sets of at least \(l+1\) vertices, each inducing a connected subgraph, with weights \(0\le a_C\le y^{|C|}\), and let \(0{\lt}xy\le 1/2\). Then (78) holds with \(\xi =1/\ln (1/(xy))\).
For real \(J\), \(S\) and \(x\) there is \(\beta _0{\gt}0\) such that \(x(e^{J\beta }-1)e^{SJ\beta }\le 1/2\) for every \(0\le \beta \le \beta _0\). This is the choice of \(\beta _0\) in [ 239 , lines 1242–1245 ] .
The left side is continuous in \(\beta \) and vanishes at \(\beta =0\).
32.30 The number of final states in Hastings’ hierarchical construction
The hierarchical construction of [ 239 ] ends with a family of final states, and the source asserts that once each of them lies within energy \(\Delta E/2\) of the ground state there are at most \(2n\) of them, \(n\) being the number of ground states [ 239 , lines 722–730 ] . The source gives no argument; the following one rests only on the gap and on Bessel’s inequality [ 164 ] .
Let \(P\) be an orthogonal projection on \(\mathbb C^d\) and let \(\phi _k\), \(k\in S\), be orthonormal. Then \(\langle \phi _k,P\phi _k\rangle =\| P\phi _k\| ^2\) and
Since \(P\) is self-adjoint and \(P^2=P\), \(\langle \phi ,P\phi \rangle =\langle \phi ,P^2\phi \rangle =\| P\phi \| ^2\). With \(e_j\) the standard basis, \(\| P\phi _k\| ^2=\sum _j|\langle \phi _k,Pe_j\rangle |^2\). Exchanging the sums and applying Bessel’s inequality to each \(Pe_j\) gives \(\sum _k\| P\phi _k\| ^2\le \sum _j\| Pe_j\| ^2=\sum _jP_{jj}=\operatorname {tr}P\).
Let \(P\) be an orthogonal projection and \(H\) an operator with \(H-E_0\ge \Delta E\, (1-P)\) for some \(\Delta E{\gt}0\), that is, \(\Delta E\, (\| v\| ^2-\langle v,Pv\rangle )\le \langle v,Hv\rangle -E_0\| v\| ^2\) for every \(v\). If \(\phi _k\), \(k\in S\), are orthonormal with \(\langle \phi _k,H\phi _k\rangle \le E_0+\theta \Delta E\), then
In particular at most \(2\operatorname {tr}P\) orthonormal states have energy at most \(E_0+\Delta E/2\). With \(P\) the projection onto the \(n\) ground states this is the count of [ 239 , lines 722–730 ] , for final states whose energy under \(H\) is within \(\Delta E/2\) of the ground energy [ 164 ] .
For each \(k\) the gap gives \(\Delta E\, (1-\| P\phi _k\| ^2)\le \theta \Delta E\), so \(\| P\phi _k\| ^2\ge 1-\theta \). Summing over \(k\) and applying Theorem 32.30.1 gives \((1-\theta )|S|\le \operatorname {tr}P\); take \(\theta =1/2\) for the second statement.
32.31 Low-energy terms that fail to commute without translation invariance
When the ground space is degenerate, the local approximations of the Hamiltonian determine the ground states only in a basis that diagonalizes their compressions to the ground space, and such a basis exists approximately when these compressions nearly commute. For translation-invariant systems they do [ 239 , lines 1603–1642 ] . The source illustrates the failure without translation invariance by an example in which the local terms \(H_i\) themselves have non-commuting compressions [ 239 , lines 1655–1676 ] , and states that for long enough filtering times and projection ranges the filtered approximations inherit this. Only the statement about the terms \(H_i\) is formalized here; the passage to the filtered approximations is not argued in the source and is not formalized [ 147 ] .
On a chain of \(m+3\) spins \(1/2\), with sites \(0,1,\ldots ,m+2\), let
and let \(H=\sum _iH_i\). The source does not specify the chain beyond site \(0\); further sites with \(H_i=S^z_i\) would only add scalars on the ground space.
The terms acting on site \(0\) cancel, and \(H=-S^z_1+\sum _{i\ge 2}S^z_i\). Its least eigenvalue is \(E_0=-(m+2)/2\), and the eigenspace for \(E_0\) is spanned by the two configurations with spin \(1\) up, every spin \(i\ge 2\) down and spin \(0\) free. In particular a vector \(v\) satisfies \(Hv=E_0v\) exactly when \(Pv=v\), for the orthogonal projection \(P\) onto these two configurations.
Sum the terms: the operators on site \(0\) add to zero. The remaining operator is diagonal in the product basis, with diagonal entry \(-s_1+\sum _{i\ge 2}s_i\) for spins \(s_i=\pm \tfrac 12\). Each summand is at least \(-\tfrac 12\), with equality exactly for \(s_1=\tfrac 12\) and \(s_i=-\tfrac 12\), \(i\ge 2\).
The source writes this Hamiltonian as \(\sum _{i\neq 0}S^z_i\); the sign of \(S^z_1\) is reversed there, which changes the ground configurations but not the dimension of the ground space or the conclusions below [ 147 ] .
The compressions \(PH_0P\), \(PH_1P\) and \(PH_2P\) to the ground space pairwise fail to commute, while \(PH_0P\) commutes with \(PH_iP\) for every \(i\ge 3\).
On the ground space \(S^z_k\) is a scalar for every \(k\neq 0\), and an operator on site \(0\) acts through its \(2\times 2\) matrix. So \(PH_0P\), \(PH_1P\) and \(PH_2P\) are, up to scalar multiples of \(P\), the compressions of \(S^x_0+S^y_0\), \(-S^x_0\) and \(-S^y_0\), and their commutators are \(\pm \) the compression of \([S^x_0,S^y_0]=iS^z_0\neq 0\). For \(i\ge 3\), \(PH_iP\) is a scalar multiple of \(P\).
The source states the last claim for \(i\ge 2\), but the pair \((0,2)\) is among the non-commuting ones [ 147 ] . The failure of commutation is localized near site \(0\): the two ground states differ only there, and only \(H_0\), \(H_1\) and \(H_2\) fail to commute on the ground space.
32.32 The trace-norm bound and every temperature in Hastings’ thermal expansion
Two steps of [ 239 , Section IV ] follow the high-temperature cluster expansion. The first turns its bound on expectation values into a bound in trace norm. The second extends the matrix product form of Section 32.21 to every temperature by taking an even power of the high-temperature form. This section proves both steps for general operators. The truncated density matrix \(\rho (\beta ,l_{\mathrm{proj}})\) of the cluster expansion enters only through the hypotheses, and the error bound at every temperature [ 239 , lines 1322–1349 ] is not proved here. The trace norm of a square matrix \(D\) is \(\| D\| _1=\operatorname {Tr}\sqrt{D^\dagger D}\), and \(\| O\| \) is the operator norm.
Let \(D\) be a square complex matrix and \(\epsilon \) a real number. If \(|\operatorname {Tr}(OD)|\le \epsilon \) for every matrix \(O\) with \(\| O\| \le 1\), then \(\| D\| _1\le \epsilon \).
The polar decomposition writes \(D=U\sqrt{D^\dagger D}\) with \(U\) unitary. Then \(O=U^\dagger \) has \(\| O\| \le 1\) and \(\operatorname {Tr}(OD)=\operatorname {Tr}\sqrt{D^\dagger D}=\| D\| _1\). For a Hermitian \(D\) this \(O\) may be taken to be the operator of signs of the eigenvalues of \(D\) that [ 239 , lines 1291–1294 ] uses; the polar decomposition removes the need for \(D\) to be Hermitian.
Let \(\rho \) and \(\sigma \) be square complex matrices with \(\operatorname {Re}\operatorname {Tr}\rho \ge 0\), and let \(\epsilon \ge 0\). If
for every matrix \(O\), then \(\| \rho -\sigma \| _1\le \epsilon \operatorname {Re}\operatorname {Tr}\rho \). For the thermal state \(\rho =\rho (\beta )\), which is positive with \(\operatorname {Tr}\rho (\beta )=Z=\| \rho (\beta )\| _1\), and \(\sigma =\rho (\beta ,l_{\mathrm{proj}})\), the hypothesis is the bound of eq. (perc2) with \(\epsilon =\exp [N\exp (-l_{\mathrm{proj}}/\xi _{\mathrm{perc}})]-1\), and the conclusion is the bound of [ 239 , lines 1281–1299 ] .
For \(\| O\| \le 1\) the hypothesis gives \(|\operatorname {Tr}(O(\rho -\sigma ))|\le \epsilon \operatorname {Re}\operatorname {Tr}\rho \), since \(\epsilon \ge 0\). Apply Theorem 32.32.1 to \(D=\rho -\sigma \).
Let \(\rho \) be a Hermitian operator in matrix product form of range \(l\), and let \(m\) be even. Then \(\rho ^m\) is positive semidefinite and in matrix product form of range \(l\). For \(\beta \ge 0\) and \(\beta _0{\gt}0\) there is an even \(m\ge 2\) with \(\beta /m\le \beta _0\). Applied to the truncated high-temperature form \(\rho (\beta /m,l_{\mathrm{proj}})\), which is Hermitian, this is the positivity and the matrix product form of \(\rho (\beta ,l_{\mathrm{proj}})=\rho (\beta /m,l_{\mathrm{proj}})^m\) in eq. (mplt) of [ 239 , lines 1303–1317 ] , with positive semidefiniteness in place of the source’s positive definiteness, which would need \(\rho (\beta /m,l_{\mathrm{proj}})\) to be invertible [ 99 ] . The count of index values in lines 1318–1320 is not established; it is the count recorded as open for Theorem 32.21.5.
For \(m=2k\), \(\rho ^m=(\rho ^k)^\dagger \rho ^k\) since \(\rho ^k\) is Hermitian, and this is positive semidefinite. Products of operators in matrix product form of range \(l\) are again in that form, with the indices at each site grouped into one (Theorem 32.21.4). For the number of factors take \(m=2(\lceil \beta /\beta _0\rceil +1)\).
32.33 The truncated approximation of a filtered term
For a term \(H_i\) of a local Hamiltonian \(H=\sum _jH_j\), [ 239 ] approximates the filtered term \(\tilde H_i^0\) by the filter of \(H_i\) under the truncated Hamiltonian \(H_{\rm loc}=\sum _{j:\, d(i,j)\le l_{proj}-R}H_j\),
[ 239 , eqs. (mid), (odef), (hpdef), lines 285–303 ] . Here the truncation keeps any family \(s\) of terms acting inside a set \(K\) of sites containing the support \(S\) of the filtered operator \(W\), and the shell consists of the omitted terms whose supports meet \(K\). The time integral of the Lieb–Robinson bound is bounded by \(T\) times its largest value, the constants are explicit, and the Gaussian weight of eq. (mid) is read as that of eq. (thd) [ 188 ] .
If \(H\) and \(W\) act on a set \(K\) of sites, then so do \(e^{iHt}We^{-iHt}\) for every \(t\) and the filtered operator \(\widetilde W\). In particular \(M_i\) acts on the sites within distance \(l_{proj}\) of \(i\) [ 239 , lines 302–304 ] .
The operators acting on \(K\) form a closed subalgebra, so they contain the exponential \(e^{iHt}\) and the products \(e^{iHt}We^{-iHt}\). The filter is the average of these products against a probability measure, and a closed subspace contains such averages.
Let \(H\) be Hermitian and \(H'\) any matrix, and write \(\tau '_u(W)=e^{iH'u}We^{-iH'u}\). If \(\| [H-H',\tau '_u(W)]\| \le \varepsilon \) for all \(u\) between \(0\) and \(t\), then
This is the comparison of \(H_i(t)\) and \(H_i^{\rm trunc}(t)\) in [ 239 , lines 1469–1482 ] .
Put \(f(s)=e^{iHs}\tau '_{t-s}(W)e^{-iHs}\), so that \(f(0)=\tau '_t(W)\) and \(f(t)=e^{iHt}We^{-iHt}\). Differentiating, \(f'(s)=e^{iHs}\, i[H-H',\tau '_{t-s}(W)]\, e^{-iHs}\), whose norm is at most \(\varepsilon \) because \(e^{iHs}\) is unitary. The mean value inequality on the interval between \(0\) and \(t\) gives the bound.
Let \(H\) and \(H'\) be Hermitian, \(q\ge 0\), \(T\ge 0\) and \(\varepsilon \ge 0\). If \(\| e^{iHt}We^{-iHt}-e^{iH't}We^{-iH't}\| \le \varepsilon \) for all \(|t|\le T\), then the filters of \(W\) under \(H\) and under \(H'\) differ in norm by at most \(\varepsilon +2\| W\| e^{-(T\Delta E)^2/(2q)}\) [ 239 , eq. (dmi) and lines 1524–1533 ] .
The difference of the filters is the Gaussian average of the difference of the evolutions. Split the average at \(|t|=T\): inside, the integrand is at most \(\varepsilon \); outside, it is at most \(2\| W\| \), and the Gaussian mass of the complement of \([-T,T]\) is at most \(e^{-(T\Delta E)^2/(2q)}\).
Under the finite-range interaction assumptions of Theorem 28.9.11, let \(A\) act on a finite set \(S\) and \(B\) on a finite set \(Y\), with \(m\le d_G(x,y)\) for all \(x\in S\) and \(y\in Y\). For every \(\mu \ge 0\),
with \(C=2+4|b_0|e^{2\mu R}\) and \(v=2v_R|b_0|e^{2\mu R}+1\).
Let \(E\) be the normalized partial trace over \(Y\). Then \(E(\tau _t(A))\) acts outside \(Y\) and commutes with \(B\), so \(\| [B,\tau _t(A)]\| \le 2\| B\| \, \| \tau _t(A)-E(\tau _t(A))\| \). By Lemma 28.9.18, the last norm is at most the sum over \(y\in Y\) of the single-site commutator bounds of Theorem 28.9.11, each at most \(C|S|\, \| A\| e^{v|t|-\mu m}\).
Let \(W\) act on \(S\subseteq K\), let \(s\) be a family of terms acting inside \(K\), put \(H_s=\sum _{k\in s}h_k\), and let \(m\) bound from below the graph distance from \(S\) to the support of every omitted term meeting \(K\). Then
the sum running over the omitted terms meeting \(K\) [ 239 , lines 1469–1505 ] .
The difference \(H-H_s\) is the sum of the omitted terms. The evolution under \(H_s\) acts on \(K\), so an omitted term whose support is disjoint from \(K\) commutes with it. For an omitted term meeting \(K\), apply Theorem 32.33.4 to the family in which the terms outside \(s\) are replaced by \(0\), which generates the evolution under \(H_s\) and satisfies the same interaction bounds, and use \(|X_k|\le v_R\).
In the setting of Theorem 32.33.5, let every term have norm at most \(J\) and let \(N\) be the number of omitted terms meeting \(K\). Let \(M\) be the filter of \(W\) under \(H_s\). Then \(M\) acts on \(K\), \(\| M\| \le \| W\| \), \(M\) is Hermitian when \(W\) is, and for every \(q\ge 0\), \(T\ge 0\) and \(\mu \ge 0\),
With \(W=H_i\), \(K\) the sites within distance \(l_{proj}\) of \(i\) and \(s\) the terms with \(d(i,j)\le l_{proj}-R\), the operator \(M\) is \(M_i\), and this is the analogue of [ 239 , eq. (apb), lines 300–320 ] with explicit constants and \(q\), \(T\) free. The comparison of the shell count with \(N(l_{proj})\) for this choice is recorded in [ 188 ] .
For \(|t|\le T\) and \(u\) between \(0\) and \(t\), Theorem 32.33.5 bounds the commutator by \(\varepsilon =2v_RC|S|\, \| W\| JN\, e^{vT-\mu m}\), so Theorem 32.33.2 bounds the difference of the evolutions by \(\varepsilon T\), and Theorem 32.33.3 gives the estimate. The support is Theorem 32.33.1, the norm bound is that of Theorem 32.4.2, and the adjoint of the filter of \(W\) is the filter of \(W^\dagger \).
In the setting of Theorem 32.33.6, let \(q\ge 0\) and \(\Delta E{\gt}0\). At \(T=q/\Delta E\) the Gaussian tail \(e^{-(T\Delta E)^2/(2q)}\) equals \(e^{-q/2}\), so
For \(q=c_1l_{proj}\Delta E\) this is the time \(T=c_1l_{proj}\) of [ 239 , lines 337–349 ] .
Put \(T=q/\Delta E\) in Theorem 32.33.6; then \((T\Delta E)^2/(2q)=q/2\) for \(q{\gt}0\), and both sides of the tail identity are \(1\) for \(q=0\).
In the setting of Theorem 32.33.6, let \(q\ge 0\) and \(\Delta E{\gt}0\), and let every eigenvalue of \(H=\sum _kh_k\) equal \(E_0\) or be at least \(E_0+\Delta E\). Write
For every ground vector \(\psi \), the component of \(M\psi \) orthogonal to the ground space has norm at most \(\delta \| \psi \| \). If \(\Omega \) is a unique ground state with gap \(\Delta E\), then \(\| M\Omega -\langle \Omega ,M\Omega \rangle \Omega \| \le \delta \). With \(W=H_i\) and the choices of Theorem 32.33.6, these are eq. (facld) of [ 239 , lines 322–348 ] and the bound for one ground state at lines 375–387, with explicit constants and \(q\) free.
Take \(\mu {\gt}0\) fixed, \(m=l_{proj}-3R\) and \(q=c_1l_{proj}\Delta E\) with \(0{\lt}c_1{\lt}\mu /v\), and suppose the number \(N\) of omitted terms meeting \(K\) grows subexponentially in \(l_{proj}\), for instance polynomially when spheres in the graph have polynomial size. Then both terms of \(\delta \) in Theorem 32.33.8 decay exponentially in \(l_{proj}\), which is the form \(JO(e^{-l_{proj}/l_{fac}})\) of eq. (facld) [ 188 ] .