Let \(L,u,v\) be natural numbers with \(u{\lt}v\) and \(v+1{\lt}L\). Place the cuts immediately after sites \(u\) and \(v\), as in [ 4 ] . Set \(T=\{ 0,\ldots ,u\} \), \(E=\{ u+1,\ldots ,v\} \) and \(U=\{ 0,\ldots ,L-1\} \setminus (T\cup E)\). For an edge \(r\) with \(r+1{\lt}L\), write \(D_r=\{ r,r+1\} \). Then \(D_u\subseteq T\cup E\), \(D_u\cap T=\{ u\} \), \(D_v\cap T=\varnothing \) and \(D_v\cap E=\{ v\} \). Every other edge has its support contained in \(T\), in \(E\), or in \(U\). Adjacent cuts, \(v=u+1\), are included.

In the setting of Definition 31.1.7, let \(E_0,\Delta \) be real with \(\Delta {\gt}0\), and suppose the original vector \(\Omega \) satisfies

Define the vectors in the paired physical coordinates by \(\Psi (p(\sigma ,\tau ))=\Omega (\sigma )\Omega (\tau )\) and \(\Phi =F\Psi \). Then

In particular, these vectors give the ground lines of the two Hamiltonians, with common ground energy \(2E_0\) and gap at least \(\Delta \). This is the contiguous-chain assertion in the paragraph “Two-sided Gaussian filtering” of  [ 292 ] , before the filtering construction; the distinction from arbitrary finite induced domains is recorded in  [ 46 ] .

Let \(W\) be a set of chain sites with \(|W|\le n\). At most \(2^n\) admissible interaction supports are contained in \(W\).

Let \(L,q,R\) be natural numbers, let \(J\) be real, and let \(H=\sum _X h_X\) be a finite-range Hamiltonian on the chain of \(L\) sites. Suppose that for every \(X\) with \(h_X\ne 0\) there is a natural number \(a\) such that \(a+R+1\le L\) and \(X=[a,a+R+1)\). For natural numbers \(l,r\), put \(A=[l,r)\cap \{ 0,\ldots ,L-1\} \). Then

This is the consecutive-window counting refinement in the remark on sharper constants of  [ 292 , Part I ] .

Let \(q\ge 1\), and suppose that every nonzero interaction is assigned to a full window of \(R+1\) consecutive sites contained in the chain. For every coordinate interval \(A\), put

Then \(\mathcal B(A)\le 1+2R\log ^2(e q^{R+1})\).

Let \(L,R\) be natural numbers, let \(q\ge 1\) be an integer, and let \(J\ge 0\) and \(\Delta {\gt}0\) be real. Let \(H=\sum _X h_X\) be a finite-range Hamiltonian on the chain of \(L\) sites. Suppose that for every \(X\) with \(h_X\ne 0\) there is a natural number \(a\) with \(a+R+1\le L\) and \(X=[a,a+R+1)\). Let \(E_0\) be real and let \(\Omega \) be a unit vector satisfying \(H\Omega =E_0\Omega \) and \(H-E_0\ge \Delta (1-|\Omega \rangle \langle \Omega |)\). For natural numbers \(l,r\), let \(\rho _A\) be the reduced state on \(A=[l,r)\cap \{ 0,\ldots ,L-1\} \), and let \(K\) take the value \(-\log p\) with probability \(p\) over the positive eigenvalues of \(\rho _A\). Put \(B=1+2R\log ^2(e\, q^{R+1})\). For every real \(u\) with \(|u|\le 1/(32\sqrt{(1+J/\Delta )B})\),

This is the marginal moment estimate with the consecutive-window count from the remark on sharper constants of [ 292 , Part I ] .

Let \(L,R,l,r\) be natural numbers. For each natural number \(i\), define the coordinate interval

Let \(Y\) be a set of chain sites with \(|x-y|\le R\) for all \(x,y\in Y\). Suppose \(Y\) meets both \(X_j\) and its complement for some \(j\ge 0\). For every natural number \(i\),

In particular, \(Y\) is split by at most one of these intervals, including \(X_0\). For a nonempty initial interval, these coordinate intervals are its metric enlargements by \(iR\), so this gives the single-splitting lemma of [ 292 , Part I ] . The coordinate statement also holds for arbitrary endpoints \(l,r\) and for \(R=0\).

Let \(R\) be a nonnegative integer and put \(N_R=2\cdot 2^{2R}\) and \(B_R(q)=1+N_R\log ^2(e q^{R+1})\). For every integer \(q\ge 2\),

Here \(N_R\) is the interval crossing bound for the family of all admissible supports, as in the cut-budget definition of [ 292 , Part I ] . The remark on sharper constants in that part describes its application to the logarithmic dependence of the area-law constant.

Let \(\Omega \) be a gapped ground vector, with gap \(\Delta {\gt}0\), of a finite-range chain Hamiltonian with local dimension \(q\ge 1\) and term norms at most \(J\ge 0\). For an interval \(X\), let \(\rho _X\) be its reduced density matrix and put

For \(0{\lt}\alpha {\lt}1\), define

Zero eigenvalues contribute zero to the trace. For every \(\alpha \in [1-u_0,1]\),

This is the near-one Rényi bound in Part I of [ 292 ] .

Let \(L,q,R\) be natural numbers, let \(J\in \mathbb R\), and let \(H=\sum _Xh_X\) be a range-\(R\) Hamiltonian on the contiguous chain \(I=\{ 0,\ldots ,L-1\} \), with local dimension \(q\) and \(\| h_X\| \le J\). Let \(D\subseteq I\) be arbitrary. In the paired coordinates \(p\) of Definition 31.1.5, put

Every \(K_{X,b}\) is Hermitian, is supported on the original set \(X\), and satisfies \(\| K_{X,b}\| \le 2J\). Let \(\mathcal C\) be the set of original supports meeting both \(D\) and \(I\setminus D\), and let \(N_R=2\cdot 2^{2R}\). For the doubled operator \(\widehat H\),

Let \(Z=\bigcup _{X\in \mathcal C}X\) be the union of crossing supports and put \(\Delta H=F\widehat H F^\dagger -\widehat H\). The operator \(\Delta H\) is supported on \(Z\). For every \(z\in Z\) there is a boundary position \(a\in \partial D\) with \(|z-a|\le R\), and

These assertions include \(q=0\), the empty chain, and empty or full \(D\). They give the boundary decomposition on a contiguous chain used in the paragraph “Two-sided Gaussian filtering” of [ 292 ] ; the distinction from arbitrary finite induced domains is recorded in  [ 46 ] .

Let \(L,q,R\) be natural numbers. Let \(K\) be a finite set of interaction labels, and for each \(k\in K\) let \(X_k\) be a nonempty support of diameter at most \(R\) in the contiguous chain \(I=\{ 0,\ldots ,L-1\} \). Let \(h_k\) be Hermitian operators supported on \(X_k\), and suppose that \(b_0\in \mathbb R\) satisfies

for every site \(x\). Put \(H=\sum _{k\in K}h_k\) and \(\tau _t(A)=e^{itH}Ae^{-itH}\), and set

These constants are independent of \(L,q\), the interaction family, and the observable support. For \(S\subseteq I\) let \(d(S,y)=\min _{x\in S}|x-y|\), with \(d(\varnothing ,y)=\infty \) and \(e^{-\mu \infty }=0\). For every operator \(A\) supported on \(S\), every site \(y\), every operator \(B_y\) supported on \(\{ y\} \), and every real \(t\),

This estimate allows every natural local dimension \(q\). If \(q\ge 1\), define, for every real \(\ell \ge 0\), \(N_\ell (S)=\{ y\in I:\exists x\in S,\ |x-y|\le \ell \} \). The normalized partial-trace expectation \(E_{N_\ell (S)}\) then satisfies

Both estimates allow \(S=\varnothing \). This is the contiguous-chain case of the lemma “Lieb–Robinson bound on a chain” in Part II of  [ 292 ] , as used for its Theorem B; the distinction from the broader finite-domain reading is recorded in  [ 46 ] .

Let \(H\) be a range-\(R\) Hamiltonian on the contiguous chain \(I=\{ 0,\ldots ,L-1\} \) with local dimension \(q\ge 1\) and interaction bound \(J\in \mathbb R\). Let \(D\subseteq I\), and let \(K=\widehat H\) and \(K'=\widehat H'\) be its doubled and regionally swapped doubled operators. Write \(\mathcal C_D\) for the original supports meeting both \(D\) and its complement, and put \(Z=\bigcup _{X\in \mathcal C_D}X\). For every real \(\ell \ge 0\) and \(t\in \mathbb R\), define

Set

Then the normalized expectation on the paired physical sites satisfies

No nonemptiness of the chain or of \(\mathcal C_D\), sign condition on \(J\), or spectral-gap hypothesis is required. This is the intermediate generator estimate in the localization paragraph of the proof of the information-reset lemma in Part II of  [ 292 ] , on a contiguous chain; the domain distinction is recorded in  [ 46 ] .

For every natural number \(R\) and real \(J\), there are positive constants \(C,v\), depending only on \(R,J\), with the following property. Let \(q\ge 1\) and \(L\) be natural numbers, let \(H\) be a range-\(R\) Hamiltonian with interaction bound \(J\) on the contiguous chain \(I=\{ 0,\ldots ,L-1\} \), let \(D\subseteq I\), and let \(\ell \ge 0\) be real. Put \(K=\widehat H\), \(K'=\widehat H'\) and \(Z=\bigcup _{X\in \mathcal C_D}X\), where \(\mathcal C_D\) is the set of original supports crossing \(D\). Write \(N(S)=N_\ell (S)\). There is a differentiable unitary path \(U\), supported on \(N(Z)\) at every real time, such that

The same path is chosen before all the following data. Let \(Q\subseteq I\), let \(W\) be any operator supported on \(Q\), and let \(\eta \ge 0\) and \(T\ge 0\) be real. Let \(\gamma _\eta \) be the centred Gaussian measure of variance \(\eta \), with \(\gamma _0\) the point mass at zero. Put \(B=N(Q\cup Z)\) and define the operators

Then \(M_{\mathrm{loc}}\) is supported on \(B\) and

These assertions include the empty chain and empty supports. This is the localized integral in the localization paragraph of the information-reset proof in Part II of  [ 292 ] , on a contiguous chain; the domain distinction is recorded in  [ 46 ] .

Fix a natural local dimension \(q\ge 1\), a natural range \(R\), a real interaction bound \(J\), a gap \(\Delta {\gt}0\), and an exponent \(p{\gt}0\). There are \(C_p{\gt}0\) and a natural number \(L_0\ge 1\), depending only on these fixed parameters, with the following property. For every \(L\ge L_0\), let \(H\) be a range-\(R\) Hamiltonian with interaction bound \(J\) on the contiguous chain \(I=\{ 0,\ldots ,L-1\} \). Let \(\Omega \) be a normalized ground vector, with ground energy \(E_0\) and full-system gap at least \(\Delta \). Let \(D,C\subseteq I\) be disjoint, let \(R_0\subseteq D\) and \(C_0\subseteq C\), and let \(b\in \mathbb R\) satisfy \(I_\Omega (D:C)\le b\). Put \(w=C_p(1+b+\log L)\) and suppose that

Then

The constants \(C_p,L_0\) precede the length, Hamiltonian, state, regions and information bound. Only the original full-system gap is assumed. This is the contiguous-chain form of the information-reset lemma in Part II of  [ 292 ] ; the domain distinction is recorded in  [ 46 ] .

Let \(I=\{ 0,\ldots ,L-1\} \) and let \(l,r,v\) be natural numbers. With truncated subtraction at the left endpoint,

Fix natural numbers \(q\ge 1\), \(R\) and \(N\), real numbers \(J\ge 0\) and \(\Delta {\gt}0\), and a real number \(B\ge 1\). Put \(\vartheta =J/\Delta \) and \(\ell _R=\log (e\, q^{R+1})\). Let \(H=\sum _{i\in F}h_i\) act on a chain of any finite length \(L\), with local spaces \(\mathbb C^q\). Here \(F\) is finite, each \(h_i\) is Hermitian, has norm at most \(J\), and is supported on a set \(S_i\) of diameter at most \(R\). Suppose that every interval \(D=[u,v)\cap I\), where \(I=\{ 0,\ldots ,L-1\} \), is split by at most \(N\) of the sets \(S_i\) and that

Choose a natural number \(m\) satisfying

Let \(\Omega \) be a unit vector satisfying \(H\Omega =E_0\Omega \) and \(H-E_0\ge \Delta (1-|\Omega \rangle \langle \Omega |)\). For natural endpoints \(l,r\) put \(X=[l,r)\cap I\) and \(T=(\lbrack l-mR,r+mR)\cap I)\setminus X\), with truncated subtraction at the left endpoint. Then

Let \(\Omega \) be a unit gapped ground vector, with gap \(\Delta {\gt}0\), of a range-\(R\) chain Hamiltonian with local dimension \(q\ge 1\) and term norms at most \(J\ge 0\). Put \(\vartheta =J/\Delta \), \(N_R=2\cdot 2^{2R}\) and \(B_R=1+N_R\log ^2(e q^{R+1})\). Define

For natural endpoints \(l,r\), let \(I=\{ 0,\ldots ,L-1\} \) and put \(X=[l,r)\cap I\) and \(T=([l-m_{\mathrm{pol}}R,r+m_{\mathrm{pol}}R)\cap I)\setminus X\), with truncated subtraction at the left endpoint. Then

These are the constants in Lemma 4.2 of [ 292 ] .

Under the hypotheses and with the constants of Theorem 31.3.7, put

Every interval \(X=[l,r)\cap I\) satisfies \(S_\Omega (X)\le C_{\mathrm{int}}^{\mathrm{pol}}\). This is the final interval-entropy estimate in [ 292 , Part I ] .

Let \(V\) be a finite set of sites and let \(\Omega \) be a unit vector on \(\bigotimes _{x\in V}\mathbb C\), with one-dimensional local spaces. For every set \(A\) of sites, \(S_\Omega (A)=0\). This is the dimension-one case of the entropy dimension bound used in [ 292 , Part I ] .

Let \(\Omega \) be a gapped ground vector, with gap \(\Delta {\gt}0\), of a finite-range Hamiltonian with term norms at most \(J\ge 0\) on the chain of \(L\) sites with local dimension \(q\ge 1\). Every interval \(X\) satisfies

with \(C_0(q,R)=\bigl(8R\log q+\log 2\bigr)\bigl(5+E_R+128B_R\bigr) +1024\, e\, B_R\), where \(E_R=\min \{ 32N_Rq^{2(R+1)},\ 64\, e\, N_R\log ^2(e\, q^{R+1})\} \), and every subset \(A\) of the chain satisfies \(S(A)\le C_{\mathrm{int}}\, |\partial A|\). Hence the constant \(C_0(q,R)(1+J/\Delta )\), chosen before \(L\), the Hamiltonian, the ground vector and the region, bounds \(S(A)/|\partial A|\) for every gapped chain. For a natural number \(N\), let \(E(N)\), \(C_1(N)=5+E(N)+128B(N)\) and \(C_0(N)=\bigl(8R\log q+\log 2\bigr)C_1(N)+1024\, e\, B(N)\) be the same nonnegative constants with \(N_R\) replaced by \(N\), where \(B(N)\) is the budget of Lemma 31.1.10; thus \(C_0(q,R)=C_0(N_R)\). This is the connected case of Theorem A; the general case is Theorem 31.5.8.

Fix the interaction range \(R\), use the coefficient \(C_0(q,R)\) of Theorem 31.4.3, and put

which is \(C_0(2,R)/(\log 2)^3\) with the energy entry \(E_R\) replaced by its logarithmic entry. For every integer \(q\ge 2\),

For every integer \(q\ge 1\), every finite-range chain Hamiltonian with term norms at most \(J\ge 0\), every gapped ground vector with gap \(\Delta {\gt}0\), and every set \(A\) of its sites,

Here \(K_R\) is fixed before the local dimension, chain length, Hamiltonian, ground vector and region. The interaction family retains the crossing count \(N_R=2\cdot 2^{2R}\). This is the fixed-range logarithmic dependence in [ 292 , Part I, Remark 6.3 ] .

Fix natural numbers \(q\ge 1\) and \(R\), and real numbers \(J\ge 0\) and \(\Delta {\gt}0\). Put \(\vartheta =J/\Delta \), \(\ell _R=\log (e\, q^{R+1})\) and define

These are the factor \(5+E_R+128B_R\) and the constant \(C_0(q,R)\) of Theorem 31.4.3 with the crossing count \(N_R\) replaced by \(2R\) and the cut budget \(B_R\) by \(B_{\mathrm{win}}\); they are fixed before the chain length, Hamiltonian, ground vector and region. Let \(L\) be any natural number and let \(H=\sum _X h_X\) be a finite-range Hamiltonian on its chain, with local dimension \(q\) and term norms at most \(J\) as in Definition 31.1.1. Suppose that every \(X\) with \(h_X\ne 0\) has the form \(X=[a,a+R+1)\) for some natural number \(a\) satisfying \(a+R+1\le L\). Let \(E_0\) be real and let \(\Omega \) be a unit vector satisfying \(H\Omega =E_0\Omega \) and \(H-E_0\ge \Delta (1-|\Omega \rangle \langle \Omega |)\). For every set \(A\) of sites,

This combines the consecutive-window count of [ 292 , Part I, Remark 6.3 ] with the constants of Theorem 31.4.3.

Put \(c=\log 2\) and fix the absolute constants

For every natural number \(R\), every integer \(q\ge 1\), every natural chain length \(L\), and every finite-range Hamiltonian \(H=\sum _X h_X\) with term norms at most \(J\ge 0\), suppose that each nonzero \(h_X\) is assigned to a full window \(X=[a,a+R+1)\) with \(a+R+1\le L\). For every unit ground vector \(\Omega \) satisfying the original global projector gap inequality with gap \(\Delta {\gt}0\), and every set \(A\) of sites,

In particular, \(C\) and \(K\) are chosen before \(q,R,L,J,\Delta ,H,\Omega ,A\). This is the polynomial range dependence in [ 292 , Part I, Remark 6.3 ] for the consecutive-window interaction family.

Let \(\Lambda \subset \mathbb Z\) be finite, with its induced path graph, and let \(q,R\in \mathbb N\). Let \((h_k)_k\) be a finite labelled family of Hermitian operators, each supported on a nonempty set \(X_k\) of graph diameter at most \(R\). Suppose \(b_0\in \mathbb R\) satisfies \(\sum _{k:x\in X_k}\| h_k\| \le b_0\) for every site \(x\). Put \(H=\sum _k h_k\) and

Let \(A\) be supported on \(S\subseteq \Lambda \) and \(B\) at \(y\in \Lambda \). Set \(\delta =\inf _{x\in S}d_\Lambda (x,y)\), with \(\inf \varnothing =\infty \) and \(e^{-\infty /2}=0\). For every real \(t\),

This extends the commutator estimate of the Lieb–Robinson lemma in [ 292 , Part II ] to arbitrary finite integer domains. No connectivity, positive local dimension, ground-state or spectral-gap hypothesis is required.

Let \(\Omega \) be a gapped ground vector, with positive gap, of a range-zero Hamiltonian on a finite chain. Then \(S_\Omega (A)=0\) for every set \(A\) of sites.

Let \(L,R\in \mathbb N\) and let \(X\subseteq \{ 0,\ldots ,L-1\} \) be nonempty, of diameter at most \(R\). If \(|X|=R+1\), then there is an integer \(a\ge 0\) such that \(a+R+1\le L\) and \(X=[a,a+R+1)\cap \mathbb Z\).

Let \(\Lambda =\{ x_0{\lt}\cdots {\lt}x_{L-1}\} \subset \mathbb Z\) be finite. Let \(\Omega \) be a unit gapped ground vector, with gap \(\Delta {\gt}0\), of an admissible Hamiltonian on \(\Lambda \). For every finite \(A\subseteq \mathbb Z\) there is a set of labels \(A'\subseteq \{ 0,\ldots ,L-1\} \) such that

Here \(\partial A'\) is the edge boundary in the chain of labels.

Let \(\Omega \) be a gapped ground vector, with gap \(\Delta {\gt}0\), of an admissible Hamiltonian with term norms at most \(J\ge 0\) on a finite set \(\Lambda \subset \mathbb Z\) with local dimension \(q\ge 1\). Every \(A\subseteq \mathbb Z\) satisfies \(S_\Omega (A\cap \Lambda )\le C_{\mathrm{int}}(q,R,J,\Delta )\, |\partial _\Lambda A|\le C_0(q,R)\, (1+J/\Delta )\, |\partial _\Lambda A|\), with the constants of Theorem 31.4.3.

Put \(c=\log 2\) and fix the positive absolute constant

For every integer \(q\ge 1\), \(R\in \mathbb N\), \(J\ge 0\) and \(\Delta {\gt}0\), let \(\Lambda \subset \mathbb Z\) be finite and let \(H=\sum _Xh_X\) be an admissible Hamiltonian of Definition 31.5.1, with local dimension \(q\), range \(R\) and term norms at most \(J\). Write \(x_0{\lt}\cdots {\lt}x_{L-1}\) for the sites of \(\Lambda \), where \(L=|\Lambda |\). Identify each interaction support with its set \(X\subseteq \{ 0,\ldots ,L-1\} \) of labels. Suppose that whenever \(h_X\ne 0\) there is an integer \(a\) such that

Let \(\Omega \) be a unit ground vector of the original \(H\), at energy \(E_0\), satisfying \(H-E_0\mathbb 1\ge \Delta (\mathbb 1-|\Omega \rangle \langle \Omega |)\). Then every finite \(A\subseteq \mathbb Z\) satisfies

The constant \(C_{\mathrm{abs}}\) is chosen before all \(q,R,J,\Delta ,\Lambda ,H,E_0,\Omega ,A\). The full-window condition (71) is the specialization in [ 292 , Part I, the remark on sharper constants ] .

Let \(I=\{ 0,\ldots ,L-1\} \) be a contiguous chain. Write \([a,z)_I=\{ x\in I:a\le x{\lt}z\} \) for natural endpoints \(a,z\), and for a real \(\rho \ge 0\) put

Then the exact interval neighborhood is

There is no condition \(z\le L\) on the original endpoint.

Let \(q\) be any natural number and let \(\Omega \) be a normalized vector on the \(L\) sites with local dimension \(q\). Suppose a real number \(c\) bounds the entropy of every interval:

For any finite index set \(s\) and natural endpoint families \(a_i,z_i\), put \(T=\bigcup _{i\in s}[a_i,z_i)_I\). Then

Empty intervals and empty families are included, and the intervals may overlap or repeat. No nonnegativity assumption on \(c\) is added, and no Hamiltonian, gap or positive local dimension is assumed. These are the interval-neighborhood and entropy steps in Part II of  [ 292 ] .

Fix a natural local dimension \(q\ge 1\), a natural range \(R\), a real interaction bound \(J\), a gap \(\Delta {\gt}0\), an exponent \(p{\gt}0\), and a natural number \(k_0\ge 1\). There are \(D_p{\gt}0\) and a natural number \(L_0\ge 1\), depending only on these fixed parameters, with the following property. For every \(L\ge L_0\), let \(H\) be a range-\(R\) Hamiltonian with interaction bound \(J\) on the contiguous chain \(I=\{ 0,\ldots ,L-1\} \), and let \(\Omega \) be a normalized ground vector with full-system gap at least \(\Delta \). Let \(T\subseteq I\) be a union of at most \(k_0\) intervals, and let \(E\subseteq I\) satisfy

Then

Empty intervals, empty \(T\) or \(E\), and overlapping interval representations are allowed. No additional disjointness assumption is imposed. The constants precede the length, Hamiltonian, state, regions and interval representation. This is the contiguous-chain form of the logarithmic-separation lemma in Part II of  [ 292 ] ; see  [ 46 ] for the domain distinction.

Let \(L,q,D\) be natural numbers and let \(\psi \) be a vector on \(L\) sites of local dimension \(q\). If every internal cut matrix, at \(0{\lt}k{\lt}L\), has rank at most \(D\), then \(\psi \) is an open-boundary matrix product state of bond dimension at most \(D\). In particular, the conclusion holds when every cut matrix has rank at most \(D\).

Let \(\psi \) be a vector on \(L\) sites of local dimension \(q\). If \(\psi \) is a scalar multiple of a rectangular open-boundary contraction with bond dimensions \(d_0,\ldots ,d_L\), then its cut matrix \(M_k\) satisfies \(\operatorname {rank}M_k\le d_k\) for \(0\le k\le L\). In particular, \(\psi \) is an open-boundary matrix product state of bond dimension at most \(D\) if and only if \(\operatorname {rank}M_k\le D\) at every internal cut \(0{\lt}k{\lt}L\). No normalization is required. The endpoint ranks are at most one; when \(L\le 1\) there are no internal rank conditions, even for \(D=0\). This is the single-cut rank observation used in the truncation-error lower bound of  [ 274 ] ; see also the sequential truncation proposition in  [ 292 , Part IIa ] .

Let \(L,q\) be natural numbers and let \(\psi \) be a vector on \(L\) sites of local dimension \(q\). Write \(r_k\) for the rank of its cut matrix at \(k\). For natural numbers \(a,d\) with \(a+d\leq L\), one has

Consequently, if \(a\leq k\leq b\leq L\) and \(r_a,r_b\leq R\), then

If, in addition, \(q\geq 1\) and \(b-a\leq 80w\) for a natural number \(w\), then

These bounds apply without a normalization or nonvanishing assumption on \(\psi \); in particular, the first three bounds also allow \(q=0\). They give the internal-cut estimate used when passing from blocks to sites in [ 292 , Part II, proof of Theorem B ] .

Let \(L,q,D\) be natural numbers and let \(\psi \) be a vector on the chain of \(L\) sites of local dimension \(q\). For each internal cut \(0{\lt}k{\lt}L\), let \(N_k=\dim \mathcal H_{[0,k)}\) and choose an ordered orthonormal basis \(b_{k,0},\ldots ,b_{k,N_k-1}\) of \(\mathcal H_{[0,k)}\). Let \(\Pi _k\) be the orthogonal projection onto the span of its first \(\min \{ D,N_k\} \) vectors, and put \(P_k=\Pi _k\otimes 1\). Define

Then \(\Phi \) is an open-boundary matrix product state with every internal bond dimension at most \(D\), and

Empty products act as the identity, so this includes \(L=0\) and \(L=1\), without a choice of basis at a boundary cut. If \(D\ge N_k\), then \(\Pi _k=1\).

This statement holds for every choice of ordered orthonormal bases. Proposition 7.1 of [ 292 , Part IIa ] is the instance in which \(\psi \) is a unit vector and each basis is an eigenbasis of the reduced state at its cut, ordered by nonincreasing eigenvalue, with any choice of eigenvectors in a repeated-eigenvalue space. That instance, with \(\epsilon _k\) identified as the sum of the discarded eigenvalues, is Theorem 31.6.6.

Let \(L,q,D\) be natural numbers and let \(\psi \) be a unit vector on the chain of \(L\) sites of local dimension \(q\). For each internal cut \(0{\lt}k{\lt}L\), let \(\rho _k\) be the reduced state of \(\psi \) on \(\mathcal H_{[0,k)}\), let \(\Pi _k\) be the projection onto the span of the eigenvectors of \(\rho _k\) belonging to its \(\min \{ D,N_k\} \) largest eigenvalues, counted with multiplicity, and let \(\epsilon _k(D)\) be the sum of the remaining eigenvalues. With \(P_k=\Pi _k\otimes 1\), the vector \(\Phi =P_1P_2\cdots P_{L-1}\psi \) is an open-boundary matrix product state with every internal bond dimension at most \(D\), and

This is Proposition 7.1 of [ 292 , Part IIa ] . Within a repeated eigenvalue the eigenvectors are taken from one fixed orthonormal eigenbasis; every other choice is covered by Theorem 31.6.5.

In the setting of Theorem 31.6.6,

Verstraete and Cirac  [ 281 ] state this bound with a factor \(2\) on the right; the form here implies theirs.

Let \(\Omega \) be a gapped ground vector, with gap \(\Delta {\gt}0\), of a finite-range Hamiltonian with term norms at most \(J\ge 0\) on a chain of \(L\ge 0\) sites with local dimension \(q\ge 1\). For any natural number \(k\), let \(X\) be the sites \(i\) with \(0\le i{\lt}k\), let \(\rho _X\) be the reduced state, and put \(s=\sqrt{(1+J/\Delta )B_R}\). For \(w\ge 0\), let \(P\) be the spectral projection of \(\rho _X\) onto eigenvalues at least \(e^{-(C_{\mathrm{int}}+w)}\). Then \(P\) is an orthogonal projection and

The constants are independent of \(L\) and \(k\).

Let \(\Omega \) be a gapped ground vector, with gap \(\Delta {\gt}0\), of a finite-range Hamiltonian with term norms at most \(J\ge 0\) on the chain of \(L\ge 1\) sites with local dimension \(q\ge 1\). Put \(s=\sqrt{(1+J/\Delta )B_R}\) and \(c_0=\sqrt{2e^{e/2}}\). For every \(0{\lt}\varepsilon \le 1\), there is a normalized open-boundary matrix product state \(\Phi \) with \(\| \Phi -\Omega \| \le \varepsilon \) and bond dimension at most

This is a supplementary accuracy form of the argument for Theorem B in [ 292 , Part IIa ] ; the stated theorem there has error \(1/L\).

Fix natural numbers \(q\ge 1\) and \(R\), and real numbers \(J\ge 0\) and \(\Delta {\gt}0\). With \(B_{\mathrm{win}}\) and \(C_{0,\mathrm{win}}\) as in Theorem 31.4.5, put

These constants are fixed before the chain length, Hamiltonian, ground vector and accuracy. Let \(H=\sum _Xh_X\) act on a chain of \(L\ge 1\) sites, with self-adjoint terms \(h_X\) of norm at most \(J\), and suppose that every nonzero \(h_X\) has support \(X=[a,a+R+1)\) with \(a+R+1\le L\). Let \(\Omega \) be a unit vector satisfying \(H\Omega =E_0\Omega \) and \(H-E_0\ge \Delta (1-|\Omega \rangle \langle \Omega |)\). For every \(0{\lt}\varepsilon \le 1\), there is a normalized open-boundary matrix product state \(\Phi \) such that \(\| \Phi -\Omega \| \le \varepsilon \) and its bond dimension is at most

This is the prescribed-accuracy consequence of the argument for Theorem B in [ 292 , Part IIa ] , with the consecutive-window constants from Part I, Remark 6.3.

Let \(\Omega \) be a gapped ground vector, with gap \(\Delta {\gt}0\), of a finite-range Hamiltonian with term norms at most \(J\ge 0\) on the chain of \(L\ge 1\) sites with local dimension \(q\ge 1\). Put \(s=\sqrt{(1+J/\Delta )B_R}\), \(c_0=\sqrt{2e^{e/2}}\) and

There is a normalized open-boundary matrix product state \(\Phi \) of bond dimension at most \(D(L)\) with \(\| \Phi -\Omega \| \le 1/L\).

Under the Hamiltonian and ground-state hypotheses of Theorem 31.3.8, suppose \(L\ge 1\). Put \(s=\sqrt{(1+J/\Delta )B_R}\) and \(c_0=\sqrt{2e^{e/2}}\). For every \(0{\lt}\varepsilon \le 1\) there is a unit open-boundary matrix product state \(\Phi \) with \(\| \Phi -\Omega \| \le \varepsilon \) and bond dimension at most

This is the prescribed-accuracy form of the proof of Theorem B in [ 292 , Part IIa ] . The source states the case \(\varepsilon =1/L\).

Under the Hamiltonian and ground-state hypotheses of Theorem 31.3.8, suppose \(L\ge 2\). Put \(s=\sqrt{(1+J/\Delta )B_R}\), \(u_0=1/(32s)\) and \(c_0=\sqrt{2e^{e/2}}\). Define

There is a unit nonzero open-boundary matrix product state \(\Phi \) of bond dimension at most \(D\) such that

This gives the normalized phase conclusion of [ 292 , Part IIa, Theorem B ] with phase zero and its printed interval constant.

Under the hypotheses of Theorem 31.6.13, for every \(0{\lt}\varepsilon \le 1\) there is a unit open-boundary matrix product state \(\Phi \) with \(\| \Phi -\Omega \| \le \varepsilon \) whose bond dimension \(D\) satisfies

The case \(\varepsilon =1/L\) is the bond bound \(L^{128s}\) of Theorem B.

The final suffix has length in \([3t,4t)\), every completed main block has at most \(t\) sites, and every remainder party has fewer than \(4t\) main sites. In particular \(d_p\le q^{4t}\) for every party. Moreover \(20m\le L\) and the number of parties is \(2m+1\le 2L\).

For every stage \(0\le k{\lt}m\), one has \((k+4)t\le L\). If \(k\ge 1\), one also has \(8w\le kt\).

Let \(\Omega \) be a unit vector on the chain. Initially place the main copy at \(R_0\) and each auxiliary copy at its specified party \(C\). There are \(1+m+\max \{ m-1,0\} \) copies, each held entirely by one party. Take a party’s private memory to be the tensor product of the copies it holds, with their product reference vector. These vectors have norm one. Private maps and changes of tensor-factor order prepare the complete copy reference exactly, with no nonprivate operation.

The coordinate map at party \(p\) depends precisely on the main configuration on \(M_p\) and the unchanged auxiliary configuration on \(\mathcal A_p\). For every terminal vector \(\Psi \), with no product or normalization assumption, its physical density after applying \(T\) and discarding all private registers is

where the expression on the right uses the original main and auxiliary sheet coordinates, and \(U_\chi \) is their specified main block change of coordinates.

The unitary \(T\) is realized by a finite sequence of private operations and changes of tensor-factor order. No nonprivate gate is introduced, and the equality holds on every input vector.

For every chain vector \(\Omega \),

If \(\| \Omega \| =1\), then \(\| \Xi \| =1\), including when \(\mathcal C=\varnothing \), and

Let \(\| \Omega \| =1\) and let \(\Psi \) belong to the specified terminal memory, with \(\| \Psi \| \le 1\). For every real \(\varepsilon \) satisfying \(\| \Psi -v_{\rm ref}\| \le \varepsilon \), the main density in the original sheet coordinates obeys

Let \(L,q\ge 1\) be integers, let \(p\in \mathbb R\), and let \(\Omega \) be a unit vector on the chain. A plain sheet has one register \(\mathbb C^q\) per site and an owner assignment \(o\). Write \(\Omega _o\) for \(\Omega \) placed on those registers. Let \(T\) be owned by \(P\), put \(E=\{ x:o(x)\ne P\} \), and suppose \(I_\Omega (T:E)\le L^{-2p}\). For any party \(Q\) define

There is a birth from \(o\) to \(o'\) consisting of private contractions and at most one normalized pair source, involving only \(P,Q\). There is also a death from \(o'\) to \(o\) using private contractions, at most one normalized pair effect and no pair source, again involving only \(P,Q\). Their maps \(\mathcal B,\mathcal D\) satisfy

The information condition for the death is evaluated in its after-death assignment \(o\). In particular, every fixed \(p{\gt}0\) gives the birth and death estimates of [ 292 , Part II ] .

Let \(L,q\ge 1\) be integers, let \(p\in \mathbb R\), and let \(\Omega \) be a unit vector on the chain. Consider two labelled copies of the sheet with owner assignments \(o_1,o_2\), a set of sites \(Y\), and a party \(P\). Put

Suppose \(I_\Omega (T:E)\le L^{-2p}\). Exchange the two owner assignments on \(Y\) and leave them unchanged outside \(Y\), obtaining \(o_1',o_2'\). There is a contraction \(\mathcal X\) between the corresponding register spaces, consisting of owner-preserving reorderings and private contractions at \(P\), with no shared pair source or effect, such that

Every register retains its party; only its sheet label changes in the reordering. The two copies remain distinct even when their owner assignments agree. For \(p{\gt}0\), this includes the interval exchange of [ 292 , Part II ] .

Suppose \(o(x)=C\) on \([a-8w,a)_L\). The retired auxiliary assignment is

Moreover \(M_8=M_0[\lbrack a,a+w)_L\to B]\), with no assumption on the original owners in \(\lbrack a,a+2w)_L\). Thus the original main ownership is restored outside the initial band; this assertion concerns the main sheet, not the retained auxiliary registers.

For the exchange identities, assume only \(w\ge 1\), \(8w\le a\), \(o(x)=C\) on \([a-8w,a)_L\), \(A\ne C\) and \(B\ne C\), and put \(Z=E(M_0,C)\cup E(U_3,C)\). The exchange information regions \(T=Z\cap Y\) and \(E=Z\setminus Y\) are exactly

The endpoint assertions do not require the three parties to be distinct; the two inequalities just stated are required for the exchange-region identities.

Assume \(o(x)=C\) on \(\lbrack a-8w,a)_L\) and \(o(x)=A\) on \([a,a+w)_L\); the parties \(A,B,C\) need not be distinct. Let \(q\ge 1\) and let \(\Omega \) be a unit vector on the original chain. Put \(\delta =L^{-12}\). Assume \(I_\Omega (T_j:E_j)\le L^{-24}\) for each row of the following table:

Here \(T,E\) are the exact exchange regions above. In particular \(E(U_0,C)=\varnothing \). All information quantities concern \(\Omega \), not an evolving approximate vector. The environments in rows \(4,5,7\) are evaluated after the corresponding death; row \(6\) uses the sheet before the birth.

Place the chosen adjacent main and auxiliary occurrences between any two fixed lists of spectator sheets. Append any further registers with vector \(\xi \), where \(\lVert \xi \rVert \le 1\). Let \(\Psi _j\) be the tensor product, in this list order, of the reference vectors of all the sheets in stage \(j\), followed by \(\xi \); each sheet reference is formed from the same original \(\Omega \). There are eight circuits \(c_j\) between these register lists with at most one nonprivate gate each, its participants contained in the set in row \(j\). Each gate expansion has at most one monomial, coefficient sum at most one, at most one normalized pair effect and at most one normalized pair source per monomial. Their reference errors satisfy

The exchange circuit \(c_3\) has no nonprivate gate. All spectator occurrences, the retired auxiliary and the further registers are retained at the stated stages and owners.

At the left boundary there is also a single-death version, for arbitrary natural \(L,w\) and parties \(A,B\). Suppose \(M\) is a sheet whose owner is \(A\) on \(T_0=\lbrack 0,w)_L\), and put \(M'=M[T_0\to B]\). For \(q\ge 1\), a unit vector \(\Omega \), and \(I_\Omega (T_0:E(M',B))\le L^{-24}\), there is a circuit from \(M\) to \(M'\) with nonprivate participants in \(\{ A,B\} \) and the same bounds one on gates, monomials, coefficient sum, pair effects and pair sources, with reference error at most \(L^{-12}\). It may be placed among the same arbitrary spectator sheets and further registers. This boundary assertion imposes neither the interior margin hypotheses nor distinctness of \(A,B\), and introduces no left auxiliary copy.

The initial sheet assigns the entire main sheet to \(R_0\), and \(M_m\) has the terminal main ownership of Definition 31.7.1. Suppose \(w\ge 1\) and \(0\le k{\lt}m\). Each birth target is held by \(R_k\) immediately before that birth. Its environment outside \(R_k\) is

For \(k{\gt}0\), the owners in \(M_k^{(2)}\) on the three intervals \(\lbrack a-8w,a)_L\), \(\lbrack a,a+w)_L\) and \(\lbrack a+w,a+2w)_L\) are respectively \(P_k,R_k,P_{k+1}\). In \(M_k^{(L)}\), the owners on \(\lbrack b-8w,b)_L\), \(\lbrack b,b+w)_L\) and \([b+w,b+2w)_L\) are respectively \(P_{k+1},R_k,R_{k+1}\). When \(m{\gt}0\), the environment at the first stage for the direct death, evaluated after that death, is \(E(M_0^{(L)},P_1)=[t,L)_L\). The four specified owner changes give \(M_k^{(R)}=M_{k+1}\). If \(L\ge 3t\), the one-band main endpoints satisfy \(\widehat M_k^{(L)}=M_k^{(L)}\) and \(\widehat M_k^{(R)}=M_k^{(R)}=M_{k+1}\).

Suppose \(w\ge 1\). Denote the right copy at stage \(k\) by \(\rho _k\), for \(0\le k{\lt}m\), and the left copy used at stage \(j+1\) by \(\lambda _j\), for \(0\le j{\lt}m-1\). The retired right auxiliary comes from the initial main sheet \(\widehat M_k^{(L)}\) and has centre \(b=(k+1)t\). It assigns \([b-3w,b-2w)_L\) to \(R_k\), \([b-2w,b-w)_L\) to \(R_{k+1}\), and the remaining sites to \(P_{k+1}\). The retired left auxiliary \(\lambda _j\) is formed from \(M_{j+1}^{(2)}\) and has centre \(c=(j+1)t\). It assigns \([c-3w,c-2w)_L\) to \(R_{j+1}\), \([c-2w,c-w)_L\) to \(P_{j+2}\), and the remaining sites to \(P_{j+1}\). These are exactly the auxiliary owner functions of Definition 31.7.1, at the same original copy labels. In particular \(\rho _j\) and \(\lambda _j\) remain distinct when their centres coincide.

For all natural \(L,w,q\), \(\mathcal F_0\) is the initial main sheet followed by the original unused copies, and \(\mathcal F_m\) is exactly the specified terminal list with all original auxiliary owners. Every copy has activation stage less than \(m\). Before a positive-stage left transfer, its selected copy is unused; after the left transfer, the current right copy is still unused. At \(k=0\) every auxiliary is unchanged by the left transfer. At \(k{\gt}0\) only \(\lambda _{k-1}\) changes in that transfer, and only \(\rho _k\) changes in the subsequent right transfer. All other occurrences retain their sheets and their relative order. For every \(k{\lt}m\), one has \(\mathcal F_k^R=\mathcal F_{k+1}\). Selecting any copy \(c\) gives a permutation of \(\mathcal F_k\) in which the selected sheet and the main sheet are first, followed by the other copies in their original order with \(c\) omitted.

The first three boundaries have respectively main sheets \(M_k,M_k^{(1)},M_k^{(2)}\), with their original ordered auxiliary family. At a positive stage, \(\mathcal B_{k,2}\) is a permutation of \(\mathcal L_{k,0}\), and \(\mathcal L_{k,8}\) is a permutation of \(\mathcal F_k^L=\mathcal B_{k,10}\). Likewise \(\mathcal B_{k,10}\) is a permutation of \(\mathcal R_{k,0}\), and \(\mathcal R_{k,8}\) is a permutation of \(\mathcal F_k^R=\mathcal B_{k,18}\). These permutations select the original copy label and preserve the order of all spectator labels; no distinctness of sheet values is required. At \(k=0\) the transition from boundary \(2\) to boundary \(3\) changes the main owner on \([0,w)_L\) from \(R_0\) to \(P_1\) and changes no auxiliary. Boundaries \(3,\ldots ,10\) are equal. For every \(k{\lt}m\), \(\mathcal B_{k,18}=\mathcal F_{k+1}\), and \(\mathcal B_{k,j}=\mathcal F_k^R\) for every \(j\ge 18\).

For every \(0\le k\le m\), \(\mathcal G_{18k}=\mathcal F_k\). For every \(k{\lt}m\) and \(0\le j\le 18\),

In particular the two consecutive lists at step \(j{\lt}18\) are \(\mathcal B_{k,j}\) and \(\mathcal B_{k,j+1}\), including the last step of every stage. The sequence starts at the original initial list and equals the terminal list for every \(n\ge 18m\). If \(m=0\), the initial and terminal lists coincide and the entire sequence is constant. If \(m=1\), there is one eighteen-step stage, with its direct first left death, seven unchanged-sheet steps and one right auxiliary copy, followed by the stationary terminal tail.

Membership in the prescribed sets implies

Consequently \(J(p)\subseteq \mathcal A(p)\) and \(|J(p)|\le 2\). The position count is

This is the position count in the sweep construction of [ 292 , Part II ] .

Suppose \(q\ge 1\), \(w\ge 1\) and \(0\le k{\lt}m\). Let \(\Omega \) be a unit vector on the original chain and put \(\delta =L^{-12}\). Assume the two before-birth bounds

At \(k=0\), assume also the after-death bound

For \(k{\gt}0\), assume each of the eight information inequalities in the table of Theorem 31.9.3, with

For every \(k\), assume the eight inequalities in that table with

In each table the auxiliary starts entirely at its specified party \(C\). All nineteen premises concern the original \(\Omega \): the boundary premise is conditional on \(k=0\), and the eight left premises on \(k{\gt}0\). In particular rows \(4,5,7\) use their after-death environments, and row \(6\) uses its before-birth environment.

Let \(\psi (\mathcal F)\) denote the tensor product of the sheet references formed from \(\Omega \) in the order of the list \(\mathcal F\). There exist circuits \(c_{k,j}:\mathcal H_{\mathcal B_{k,j}}\to \mathcal H_{\mathcal B_{k,j+1}}\), for \(0\le j{\lt}18\), each with at most one nonprivate gate, whose participants lie in \(\mathcal S_{k,j}\), and

Each original nonprivate gate has at most one recorded monomial, absolute coefficient sum at most one, and at most one normalized pair source and one normalized pair effect per monomial. At stage zero, position \(2\) joins the boundaries of the direct death into \(P_1\); positions \(3,\ldots ,9\) are identity circuits, with zero error and no nonprivate gate. Every original auxiliary label and the order of all spectator occurrences are retained.

In Definition 31.11.4, let \(N(C)\) count nonprivate occurrences and let \(n_p(C)\) count those involving party \(p\). Then

Every original nonprivate gate retains the recorded monomial, coefficient-sum, pair-effect and pair-source bounds one. If \(\| \Omega \| =1\), then, with \(\Psi =C\psi (\mathcal F_0)\),

The tail \(d_n\) for \(n\ge 18m\) is the identity. When a first stage exists, its positions \(3,\ldots ,9\) are identities. If \(m=0\), the middle is the identity on the common initial and terminal list. If \(m=1\), it has one eighteen-position stage with the direct left-death boundaries, seven identities and one right auxiliary copy.

Write \(\operatorname {part}(g)\) for the participant set of a gate. Every pair of parties in one \(\mathcal S_{k,j}\) has level distance at most two, and each finite subset of \(\mathcal S_{k,j}\) has at most eight elements. Each nonprivate gate \(g\) of the middle circuit has a stage and position with

Consequently \(|\operatorname {part}(g)|\le 40\), and \(|\ell (p)-\ell (q)|\le 2\) whenever \(p,q\in \operatorname {part}(g)\).

Let \(W\) be an original circuit on these parties such that any two participants of one nonprivate gate have level distance at most two. Consider a tagged distributed link family on these parties. Suppose each party is incident to at most \(d\) links and that the two endpoints of every link participate in a common original nonprivate gate of \(W\). Then, for every natural prefix index \(r\),

The endpoint prefixes \(r=0\) and \(r{\gt}m\) have no crossing links. For the middle circuit \(C\), its derived support supplies the required gate distance. Thus the common-original-gate and degree conclusions of distributed compression at \(d=3120\) imply

Here \(3120=2\cdot 40\cdot 39\) is the degree bound in Theorem 45.48.19 at \(b=40\), from [ 287 , Theorem 5.2 ] . This gives a coarse bound for the routing argument of [ 292 , Part II, proof of Theorem B ] .

Let \(L,w,q\) be natural numbers with \(L{\gt}0\) and \(q\ge 1\), and let \(\Omega \) be a unit vector on the original chain. Suppose the circuits \(c_{k,j}\) satisfy exactly the local conditions in Definition 31.11.4: they are bounded changes between the prescribed boundary references, with participants in \(\mathcal S_{k,j}\), bounds \(r=K=1\), error \(4L^{-12}\) and pair-source bound one; the seven first-stage padding positions are identities with zero error and zero nonprivate count. Put \(N=18m\) and \(\varepsilon =4NL^{-12}\).

Let \(\mathcal F_0\) be the original ordered initial sheet list. Its sheets have constant owner; let \(h(F)\) be the owner of site zero in sheet \(F\). For each party \(a\), define \(H_a\) as the tensor product, in the original order, of the sheet spaces with \(h(F)=a\), and let \(x_a\) be the product of their \(\Omega \)-reference vectors. Then \(\| x_a\| =1\), including an empty product. From the constructed product input \(X=\bigotimes _a x_a\) there is an original circuit \(C\) to the terminal physical factors followed by all retained private registers. The physical factor at \(a\) has dimension \(d_a=q^{|M_a|}\). Its output \(\Psi =CX\) satisfies

Each original nonprivate gate has at most one recorded monomial, absolute coefficient sum at most one, and at most one normalized pair source and one normalized pair effect per monomial. Every nonprivate gate \(g\) of this same \(C\) has a stage and position with

In particular its arity is at most \(40\) and its participant levels have pairwise distance at most two.

Let \(J\) be the prescribed identification of the terminal sheet stack with its raw register list, and let \(T\) be the private terminal regrouping. Put \(Y=J^{-1}T^{-1}\Psi \). With the sheet readout \(\mathcal R\) and copy-labelled vector \(\Xi \) of Definition 31.7.8, one has

The prescribed owner-grouping isometry \(A_{\rm aux}\) identifies the private memory with the Euclidean space of the original auxiliary copy-site configurations, so that memory is finite-dimensional. Write \(V\) for the readout of \(\Psi \) in the prescribed physical coordinates and the fixed standard private orthonormal basis, and put \(\rho =\operatorname {Tr}_{\rm private}|V\rangle \langle V|\). For the fixed main-block coordinate map \(U_\chi \),

All auxiliary copy-site labels remain at their specified private owners. The vector \(\Xi \) is the product over original copies, including the scalar \(1\) for an empty copy family; it need not factor over its final parties. This is the conditional preparation step in [ 292 , Part II ] , with [ 79 ] .

For arbitrary natural \(L,w\), any finite party set \(A\) contained in one prescribed \(\mathcal S_{k,j}\) satisfies \(|A|\le 3\). Consequently every nonprivate gate of the circuit constructed in Theorem 31.12.1 has at most three parties.

Let \(\sigma \) be the distributed operator contraction with input and output alphabet \(X_a\) at each party, all finite. Retain its tagged link alphabets \(D_e\) and endpoint tensors \(A_a\). Let \(\omega \in \bigotimes _a\mathbb C^{X_a}\) be a unit vector and suppose \(\| \sigma -|\omega \rangle \langle \omega |\| _1\le \epsilon {\lt}1\). There exist one product-basis input \(z\) and one nonzero column \(v=\sigma |z\rangle \). With the same identity bonds, its coefficients are

For \(\widehat v=v/\| v\| \), one has \(\| \widehat v\| =1\) and some real \(\theta \) satisfies \(\| \widehat v-e^{i\theta }\omega \| \le 2\epsilon \). The same \(z,v\) satisfy, for every party subset \(S\),

The cut product contains one alphabet for each tagged crossing edge, retaining parallel links and allowing empty alphabets. The column selection is that of [ 287 , Lemma 8.1 ] .

Let \(i\) range over an arbitrary index set \(I\), with sizes \(s_i\ge 2\) and natural lengths and widths \(L_i,w_i\). Let \(\mathcal C_i\) be given distributed constructions on the parties of these chains, with no messages in their recorded monomial expansions; write \(W_i\) for their original circuits and \(\rho _i\) for their unrenormalized physical densities. The index map \(i\mapsto L_i\) need not be injective. Fix natural \(a,b,\chi \) with \(a\le b\) and \(8(2b(a-1))\le \chi \). Fix natural \(n,d,r,u,v\) and real \(C_N,C_D,C_K,C_S\), independent of \(i\). Subtraction in \(a-1\) is truncated at zero. For each member, all parties \(j\) and all original nonprivate gates \(g\) satisfy the following resource bounds:

For each \(i\) there is a natural number \(K_i\) bounding the number of recorded monomials at every original nonprivate gate, with \(K_i\le C_Ks_i^u\). Every recorded monomial has at most \(r\) pair effects, and at every original nonprivate gate the moduli of the recorded coefficients sum to at most \(C_Ss_i^v\). Let \(\omega _i\) be unit vectors in the physical output spaces, and let \(\eta _i\in \mathbb R\) satisfy

Assume also that any two participants of one original nonprivate gate of \(W_i\) have level distance at most two. These hypotheses concern every member of the family. For circuits constructed in Theorem 31.12.1, the geometric conclusion follows from its derived stage support.

For every real \(p{\gt}0\), there exist \(C_p\ge 0\) and an integer \(\alpha {\gt}0\), chosen before all \(i\), such that every member has a tagged distributed operator network \(\sigma _i\) satisfying

The alphabet bound in (140) holds for every tagged link \(e\). Every party meets at most \(2b(a-1)\) links, and the two endpoints of each link participate in one common original nonprivate gate of this same \(W_i\). Consequently every level prefix has at most \(\chi \) tagged crossing links. Set \(B_i=\max \{ 1,\lceil C_ps_i^\alpha \rceil _+\} \), a natural number. Every star or sample alphabet has size at most \(B_i\).

Only for a member with \(\epsilon _i{\lt}1\), choose one product-basis input \(z_i\) and a nonzero column \(v_i=\sigma _i|z_i\rangle \). This is the same input-fixed endpoint network, with \(\| \widehat v_i\| =1\) and \(\| \widehat v_i-e^{i\theta _i}\omega _i\| \le 2\epsilon _i\) for some real \(\theta _i\). For every natural prefix index \(t\), put \(S_t=\{ j:\ell (j){\lt}t\} \). The same column satisfies

The constants precede every family member, its network, the member’s condition \(\epsilon _i{\lt}1\), the useful column and all cuts. This is the conditional compression and column step of [ 287 , Theorem 5.2, Remark 5.3 and Lemma 8.1 ] .

Let \(L,w,q\) be arbitrary natural numbers, with the same owner and block-coordinate definitions, including the zero-width convention. Let \(\psi \) be any vector on the physical party factors \(\bigotimes _a\mathbb C^{d_a}\), where \(d_a=q^{|M_a|}\). Define the original-site vector exactly by

For every natural \(a\), put \(S_a=\{ b:\ell (b){\lt}a\} \). If \(a\le m\), then

The last prefix \(S_{m+1}\) includes every party, including \(R_m\), and

Scalar physical factors remain in these products. The prefix \(S_m\) ends at site \(mt\); the final remainder is included by \(S_{m+1}\). These are the original block coordinates in [ 292 , Part II, proof of Theorem B ] .

In (143), suppose \(q\ge 1\), \(w\ge 1\), \(L\ge 60w\) and \(R\in \mathbb N\) with \(R\ge 1\). Suppose the physical party tensor has the prefix bounds

for every integer \(1\le a\le m\). Then every original-site cut \(0\le k\le L\) satisfies

The vector \(\psi \) is arbitrary; no norm or nonzero premise is needed.

Let \(L,a,w,q\) be natural numbers, \(w\ge 1\) and \(8w\le a\). Let the original main owner be \(C\) on \(\lbrack a-8w,a)_L\), \(A\) on \(\lbrack a,a+w)_L\) and \(B\) on \([a+w,a+2w)_L\), with \(A\ne C\) and \(B\ne C\). No owner is prescribed outside these intervals, and \(A=B\) is allowed. For all eight pairs \((T_j,E_j)\) in the table of Theorem 31.9.3,

In particular the exchange uses \(E=Z\setminus Y\) and not the complement of \(Y\); the death rows use their after-death sheets.

Fix \(q\ge 1\), \(R\in \mathbb N\), \(J\in \mathbb R\) and \(\Delta {\gt}0\). There are \(D_{24}{\gt}0\) and \(L_0\ge 1\), depending only on these fixed parameters, such that the following holds for every \(L\ge L_0\), every Hamiltonian of range \(R\) and strength bound \(J\) with gapped ground vector \(\Omega \) and energy \(E_0\), every choice of parties, and all the ownership data just stated. If \(D_{24}\log L\le w\), then

The choice of \(D_{24},L_0\) precedes all chain, vector, width and ownership data. This is the logarithmic-separation argument for [ 292 , Part II, the band dissolution lemma ] , with the stated lattice convention and exponent \(24\).

For arbitrary natural \(L,q\), let \(w\ge 1\) and \(0\le k{\lt}m\). Put \(a=kt\) and \(b=(k+1)t\), where \(t=20w\). The target and environment pairs are

At \(k=0\), the direct death has target \([0,w)_L\) and after-death environment \(E(M_0^{(L)},P_1)=[t,L)_L\). Every such target is separated from its stated environment by at least \(w\).

For fixed \(q\ge 1\), \(R\in \mathbb N\), \(J\in \mathbb R\) and \(\Delta {\gt}0\), there are \(D_{24}{\gt}0\) and \(L_0\ge 1\) before all chain, Hamiltonian, ground-vector, width and stage data, such that \(L\ge L_0\) and \(w\ge D_{24}\log L\), with \(w\ge 1\), imply \(I_\Omega (T:E)\le L^{-24}\) for both birth pairs and, at \(k=0\), for the direct-death pair. Here \(\Omega \) is any gapped ground vector of the given Hamiltonian. These are the three main-sheet estimates in [ 292 , Part II, proof of the sweep proposition ] .

Fix \(q\ge 1\), \(R\in \mathbb N\), \(J\in \mathbb R\) and \(\Delta {\gt}0\). There are \(D_{24}{\gt}0\) and \(L_0\ge 1\) such that, for every \(L\ge L_0\), every corresponding gapped ground vector \(\Omega \) and every natural \(w\ge 1\) with \(D_{24}\log L\le w\), each stage \(k{\lt}m\) admits eighteen circuits between its prescribed successive boundaries. For each \(0\le j{\lt}18\), \(c_{k,j}\) is a bounded change on \(\mathcal S_{k,j}\) with bounds \(r=K=1\), pair-source bound one, at most one nonprivate occurrence, and reference error at most \(4L^{-12}\). At \(k=0\) the positions \(3,\ldots ,9\) are the identity circuits, have zero error and have no nonprivate occurrence. Thus these circuits satisfy the entire local conclusion of Theorem 31.11.3, including the original copy labels and spectator order. No information estimate or circuit is an additional physical premise. This gives the local circuits of [ 292 , Part II, the sweep construction and the sweep proposition ] .

Fix \(q\ge 1\), \(R\in \mathbb N\), \(J\in \mathbb R\) and \(\Delta {\gt}0\). There are \(D\ge 1\), \(K{\gt}0\), \(s\in \mathbb N\) and \(L_0\ge 1\), before all further physical data, with the following property. For every \(L\ge L_0\), Hamiltonian of these parameters, gapped ground vector \(\Omega \) with energy \(E_0\), put \(w=\lceil D\log L\rceil _+\) and \(N=18m\). The constructed initial party vectors \(x_p\) have norm one. Their product input \(X=\bigotimes _p x_p\) admits a circuit \(C\) to the specified terminal physical factors and all retained private auxiliary registers, with output \(\Psi =CX\). For every party \(p\) the same \(C\) has the stated lifetime bound. Every original nonprivate gate \(g\) has a stage \(k{\lt}m\) and position \(j{\lt}18\) such that

Every original nonprivate gate has at most one recorded monomial, absolute coefficient sum at most one, and at most one normalized pair effect and one normalized pair source per monomial. In particular its arity is also at most \(40\), and any two participants have level distance at most two. The geometric and dimension conclusions are

Use the inverse terminal regrouping to form \(Y\), the sheet readout \(\mathcal R\), and the standard physical/private readout \(V\) as in Theorem 31.12.1. Put \(\rho =\operatorname {Tr}_{\rm private}|V\rangle \langle V|\) and \(\varepsilon =4NL^{-12}\). Then

The vector \(\Xi \) is the product over the distinct original copies, including the empty product. It need not be a product over parties. Only the main factors have dimensions \(d_p\); all auxiliary registers remain private at their owners. This is the preparation assertion of [ 292 , Part II, the sweep proposition and the density estimate in the proof of Theorem B ] , with [ 79 ] .

For fixed \(q\ge 1\), \(R\in \mathbb N\), \(J\in \mathbb R\) and \(\Delta {\gt}0\), there are \(D\ge 1\), \(K{\gt}0\), \(s\in \mathbb N\) and \(L_0\ge 2\) before every physical instance. Let \(\iota \) consist of all tuples \((L,H,\Omega ,E_0,\mathrm{pos})\) with \(L\ge L_0\), \(H\) a Hamiltonian of the fixed parameters and \(\Omega \) a gapped ground vector at energy \(E_0\). For every \(i\in \iota \), set \(w_i=\lceil D\log L_i\rceil _+\). One can choose the unit initial vectors and the circuit \(C_i\) together, obtaining the construction of Definition 31.13.5 for all \(i\). With \(\omega _i=U_{\chi _i}\Omega _i\), these same constructions satisfy

Every gate has monomial bound one, every recorded monomial has at most one pair effect, and the coefficient moduli at each gate sum to at most one. Its original participants have pairwise level distance at most two. The target density is the coordinate reindexing of \(|\Omega _i\rangle \langle \Omega _i|\) by \(\chi _i\). Thus all premises of Theorem 31.12.4 hold with arity \(3\), lifetime \(40\), crossing parameter \(1600\), size \(L_i\), gate-count parameters \((n,C_N)=(1,2)\), physical-dimension parameters \((d,C_D)=(s,K)\), \(r=1\), \(u=v=0\), \(C_K=C_S=1\) and \(\eta _i=52L_i^{-11}\), using the uniform compression theorem of [ 287 , Theorem 5.2 ] . No one Hamiltonian or ground vector is selected to represent a length.

Let \(q\ge 1\), \(D\ge 1\), \(C_p\ge 0\), a natural \(\chi \), and \(\alpha \in \mathbb N\) with \(\alpha {\gt}0\). For \(L\ge 2\) and \(w=\lceil D\log L\rceil _+\), put \(A_L=\max \{ 1,\lceil C_pL^\alpha \rceil _+\} \) and \(n=\max \{ 1,\lceil 40D\log q\rceil _++\chi \alpha \} \). Then

Both the exponent and the prefactor are independent of \(L\). The inequality includes \(q=1\), \(C_p=0\) and \(\chi =0\).

Retain \(q,D,C_p,\alpha ,\chi ,L,w,A_L,n\) from Theorem 31.14.1 and assume \(60w\le L\). Let \(\psi \) be a unit vector in the physical party space. Suppose its coefficient flattening satisfies \(\operatorname {rank}F_{S_r}(\psi )\le A_L^{\chi }\) for every \(1\le r\le m\), where \(S_r=\{ p:\ell (p){\lt}r\} \). Write \(\chi _{\rm cfg}\) for the fixed main-configuration equivalence. Define the inverse main-coordinate vector \(\Phi (x)=\psi (\chi _{\rm cfg}(x))\) and \(B=q^{40w}A_L^{\chi }\). Then \(\| \Phi \| =1\), every original-site cut \(0\le k\le L\) satisfies \(\operatorname {rank}M_k(\Phi )\le B\), and \(\Phi \) is an open-boundary MPS of bond \(B\). The polynomial bound (156) holds and \(n\ge 1\). The prefix-rank bounds are hypotheses here; for the protocol vector they are proved in Theorem 31.14.3. It is the block-to-site passage in [ 292 , Part II, end of the proof of Theorem B ] .

Fix \(q\in \mathbb N\) with \(q\ge 1\), \(R\in \mathbb N\), \(J\in \mathbb R\) and \(\Delta {\gt}0\). There are \(C{\gt}0\) and \(n\in \mathbb N\) with \(n{\gt}0\), depending only on these fixed parameters, such that the following holds for every \(L\ge 2\), every Hamiltonian of range \(R\) and strength bound \(J\), and every gapped ground vector \(\Omega \) at energy \(E_0\). There are \(B\in \mathbb N\) and a vector \(\Phi \) on the original chain with

such that \(\Phi \) is an open-boundary MPS of bond \(B\), and some \(\theta \in \mathbb R\) satisfies

This is [ 292 , Theorem B ] , with a unit approximant and a positive natural polynomial exponent. Neither information estimates, circuit data nor rank bounds are hypotheses of this theorem.

Consider a family on the chain parties with the polynomial count, physical-dimension, monomial and coefficient-sum hypotheses of Theorem 45.48.17. Suppose its original circuits have arity at most three and lifetime at most forty. Let \(\omega _i\) be unit physical vectors with \(\| \rho _i-|\omega _i\rangle \langle \omega _i|\| _1\le \eta _i\). Suppose any two participants of one original nonprivate gate have placement distance at most two. The real size \(L_i\ge 2\) may be independent of the natural chain length and width used to specify the party set.

For each \(p{\gt}0\) there are \(C_p\ge 0\) and natural \(\alpha {\gt}0\), before all members, and for every member one sampled network with both density estimates, retained trace bounds, polynomial alphabets and common-original-gate witnesses in (368)–(369). On this same tagged network every party has degree at most \(160\), and every natural prefix index \(r\) satisfies

Here \(\partial _r\) consists of the links crossing \(\{ j:\ell (j){\lt}r\} \). These are auxiliary preparation hypotheses for the crossing step in [ 292 , Part II, proof of Theorem B ] .

Fix \(q\in \mathbb N\) with \(q\ge 1\), \(R\in \mathbb N\), \(J\in \mathbb R\) and \(\Delta {\gt}0\). There are \(D\ge 1\), \(K{\gt}0\), \(s\in \mathbb N\) and \(L_0\ge 2\) before all physical instances. Let \(\iota \) be the entire set of admissible tuples \((L,H,\Omega ,E_0,\mathrm{pos})\) with \(L\ge L_0\), Hamiltonian parameters \(q,R,J\), and gapped ground vector \(\Omega \) of gap \(\Delta \) at energy \(E_0\). Set \(w_i=\lceil D\log L_i\rceil _+\). There are unit initial party vectors and circuits \(C_i\) for all \(i\in \iota \) giving the constructions of Definition 31.13.5 and retaining every resource, target-density and width conclusion of Theorem 31.13.6. Their arity is at most three. In particular, for the same original circuits,

where \(\omega _i=U_{\chi _i}\Omega _i\) is the unit main-coordinate vector. Monomial and coefficient-sum bounds are one, original-gate participant distances are at most two, and \(w_i\ge 1\), \(60w_i\le L_i\).

There are \(C_p\ge 0\) and natural \(\alpha {\gt}0\), before every member of this entire family, such that each member has one sampled network \(\sigma _i\) satisfying

The incidence and prefix bounds hold at every party and every natural prefix index of this same network. Every link’s endpoints participate in a common original gate of \(C_i\). The input, circuit, physical coordinate map and unnormalized density are those of the constructed preparation. Only the main registers are physical: every auxiliary copy remains private at its specified retained copy-site owner, as in [ 79 ] . No information, circuit, arity or crossing bound is an additional physical premise. This supplies the sampled-network crossing estimate in [ 292 , Part II, proof of Theorem B ] .

The vectors \(e_i=|w_i\rangle \) are a real orthonormal family, as are \(|w_i\rangle |w_i\rangle \). Thus the matrix \(B\) with columns \(e_i\) satisfies \(B^\dagger B=1\) and \(B^\dagger =B^{\mathsf T}\).

The weights \(\lambda _i\) are nonnegative and sum to one, and \(\| \psi _{2N}(p)\| =1\). In particular, \(\| \psi _{2N}(1/N)\| =1\).

Numbering the sites in each half from zero identifies \(C\) with the coefficient matrix of the actual chain cut after its first \(N\) sites. Under the same relabelling on the left half, \(\rho \) is the regional state of these first \(N\) sites.

With \(B\) as above, \(C=B\operatorname {diag}(\sqrt{\lambda _i})B^\dagger \).

The left marginal satisfies \(\rho =B\operatorname {diag}(\lambda _i)B^\dagger \).

The characteristic polynomial of the middle marginal, and therefore of the actual first-\(N\)-site regional state, is

Thus its squared Schmidt coefficients are \(1-p\), one copy of \(p/2^N\) for each binary word, and the remaining zeros. This statement includes zero weights at \(p=0\) and \(p=1\), and applies to \(p=1/N\).

Let \(N{\gt}0\), \(0\le p\le 1\), and let \(A\) be a proper subset of the \(2N\) sites. If \(\rho _A(p)\) is the reduced state of \(|\psi _{2N}(p)\rangle \) on \(A\), then

Let \(N{\gt}0\), \(0\le p\le 1\), and let \(A\) be any subset of the \(2N\) sites. Then

The empty and full regions both have entropy zero. This bounds the entropy of the state in  [ 274 ] without a connectedness assumption on \(A\).

For \(N{\gt}0\) and every region \(A\), the source’s choice \(p_N=1/N\) satisfies \(S(\rho _A(p_N))\le 2\log 2\), or two bits.