The three-lines theorem on the strips between real parts \(0\) and \(\pm 1/4\) gives \(|f(z)|\le c\, m^{4|\operatorname {Re}z|}\le \mathrm ec\). Put \(k(z)=f(z)+f(-z)-2f(0)\), so that \(k(0)=k'(0)=0\) and \(|k|\le 4\mathrm ec\) on the disk of radius \(\delta =1/(8\ell )\). The function \(k(z)/z^2\) is entire, and the maximum principle bounds it by \(4\mathrm ec/\delta ^2\) on the disk. Since \(\operatorname {Re}f(-x)=\operatorname {Re}f(x)\), the quantity \(\operatorname {Re}(f(x)-f(0))\) equals \(\operatorname {Re}k(x)/2\). Only the real part of the linear Taylor coefficient vanishes.

The recurrence gives \(L(u)\le 2L(u/2)+2K_0u^2\) because \(-\log (1-v)\le 2v\) for \(0\le v\le 1/2\). Iterating, \(2^jL(u/2^j)\to uS\), and the geometric series of quadratic terms sums to at most \(4K_0u^2\). The tail is exponential Markov at \(u=\pm r\).

On the imaginary axis \(f\) is an expectation in a vector with the same norm. For a product, write \(\Theta _j=\sqrt{s_j}\, \omega _j\) with orthonormal \(\omega _j\). The weights satisfy \(|s_j^{1/2+z}s_k^{1/2-z}|\le s_j+s_k\), and Cauchy–Schwarz with Bessel’s inequality bounds every row and column sum of \(|c_{jk}\langle \omega _j,d\, \omega _k\rangle |\) by \(\lVert c\rVert \lVert d\rVert \). The last claim is Lemma 21.1.1 with \(m=2N\).

Summing the value at \(z_u\) over all terms gives \(\langle F^2\Psi ,\mathsf H\Psi \rangle /M(u)=E_0\). The excitation energy is therefore the sum of \(\operatorname {Re}(f_i(0)-f_i(z_u))\) over the crossing terms, each bounded by Lemma 21.2.1 since \(|z_u|\le |u|\). The gap inequality \(\mathsf H-E_0\ge g_0(1-|\Psi \rangle \langle \Psi |)\) then bounds \(1-|\langle \Psi ,\Theta \rangle |^2\).

Conjugating by \(U^*\otimes 1\), where \(U\) diagonalizes \(\rho \), preserves one-sided terms, product decompositions with their factor norms, the eigenvalue equation and the gap inequality, and makes the marginal diagonal. On the admissible interval every halving stays admissible, \(|u|\le 1/32\), \(\log (\mathrm ed_i)\le \sqrt{\mathcal B}\) and \(128\mathrm e\vartheta \mathcal Bu^2\le 1/2\), so Lemmas 21.3.1 and 21.1.2 give the moment bound. At \(u=\pm 1/(32\sqrt{(1+\vartheta )\mathcal B})\) the quadratic term is at most \(\mathrm e/2\), which gives the tail.

The factor \(c_\alpha \) is a matrix unit on \(D\cap B\) tensored with the identity on \(B\setminus D\), written as \(S_aS_b^*\) for isometries \(S_a\). The factor \(d_\alpha \) is the compression \(T_a^*hT_b\) along the isometries \(T_a\colon \psi \mapsto e_a\otimes \psi \). The identity \(h=\sum c_\alpha \otimes d_\alpha \) follows entrywise from the support property, and there are \((\prod _{v\in D\cap B}n_v)^2\le d^2\) terms.

Pass to cut coordinates. The operator norm, Hermiticity, the eigenvalue equation and the gap inequality are preserved by the relabelling. Lemma 21.4.1 supplies the one-sided terms and the product decompositions of the crossing terms. Every local dimension is positive, since \(\Psi \ne 0\). Apply Theorem 21.3.2.

The indices outside \(E\) with positive weight are exactly those in the tail event. The weight bounds are the definition of \(E\).

With \(A=U\operatorname {diag}(a)U^*\) and \(B=V\operatorname {diag}(b)V^*\), \(\operatorname{Tr}(AB)=\sum _{ij}a_ib_j|W_{ij}|^2\) for the unitary \(W=U^*V\), whose squared moduli are doubly stochastic. Apply the scalar Hölder inequality to \(a_i|W_{ij}|^{2/p}\) and \(b_j|W_{ij}|^{2/q}\).

Apply Lemma 21.5.1 with exponents \(1/(1-a)\) and \(1/a\) to \(\rho \) and \(L^2\), then the moment bound at \(u=-a/(1-a)\), and use \(1/(1-a)\le 2\).

A maximum with a common floor contracts differences of logarithms. For the cost, write \(p=\mathrm e^{m+z}\), \(q=\mathrm e^{m-z}\), so that \(p+q=2\sqrt{pq}\cosh z\) and \(|t|\le a|z|\le |z|/2\). Then \(\cosh t-1\le \frac{t^2}2\cosh t\le \frac{a^2z^2}2\cosh (z/2)\), using \(\sinh v\le v\cosh v\), and \(z^2\cosh (z/2)\le \frac{32}{\mathrm e^2}\mathrm e^{|z|}\le 8\cosh z\).

Hermitian pairing writes the real part as \(-\sum _{jk}(\cosh \log (l_j/l_k)-1)\operatorname {Re}T_{jk}\). For a product term \(|T_{jk}|=\sqrt{s_js_k}\, |c_{jk}||\langle \omega _j,d\, \omega _k\rangle |\), Lemma 21.5.3 bounds the hyperbolic coefficient times \(\sqrt{s_js_k}\) by \(2a^2(s_j+s_k)\), and the row and column estimate of Lemma 21.2.1 gives the factor \(2\lVert c\rVert \lVert d\rVert \).

The algebraic properties hold for \(K\otimes 1\) and survive the relabelling. The regional state is the partial trace of a positive semidefinite matrix, and is therefore Hermitian. The expectation formula is the defining property of the partial trace, and the last identity follows from cyclicity of the trace, since \(K^{-1}\rho _{\varphi ,D}K=\rho _{\varphi ,D}\).

Let \(u',v'\) be configurations on \(D'\). By Lemma 13.8.44, the bijection of complementary configurations changes variables in the defining partial trace:

This is the asserted conjugation identity.

By Theorem 13.6.79, conjugation by a permutation isometry preserves the von Neumann entropy of the regional state.

Differentiating along \(\mathrm e^{tB}K\mathrm e^{-tB}\) with \(B^*=-B\) gives \(\operatorname {Re}\langle \psi ,A(BK-KB)w'\rangle =0\). Since \(BK-KB=(B-KBK^{-1})K\), the descending trace argument of Lemma 21.5.5 removes the outer filters, and the expectation of \(B\) is imaginary. Hence \(\operatorname {Re}\operatorname{Tr}(K^{-1}\rho KB)=0\) for every skew-Hermitian \(B\), so \(K^{-1}\rho K\) is Hermitian. In the eigenbasis of \(K\) this forces \(\rho K^2=K^2\rho \) and then \(\rho K=K\rho \).

The joint eigenspaces form an orthogonal internal direct sum; collect orthonormal bases of the nonzero ones.

Moving mass from coordinate \(i\) to coordinate \(k\) is feasible for small \(t{\gt}0\). Its first-order effect on the squared output norm is \(a\operatorname {Re}\operatorname{Tr}\rho G\) with \(G=U\operatorname {diag}(v/x)U^*\), after the descending trace argument, and equals \(a(q_k/x_k-q_i/x_i)\). The floor-constrained optimality condition and the contraction of a common floor give the rest.

Transport by \(U^*\otimes 1\) to the Schmidt coordinates and apply Lemma 21.5.4. For disjoint supports, both operators are of the forms \(A\otimes 1\) and \(1\otimes B\) across the same cut.

Unitary conjugation preserves feasibility, so Lemma 21.5.8 gives commutation. In a common eigenbasis the eigenvalues of \(K\) are \(x^{a/2}\) with \(x\ge f\), \(\sum x=1\), by comparing \(\operatorname{Tr}K^{2/a}\) in the two bases, and diagonal variations in that basis are feasible. Lemma 21.5.10 applies with \(\sum q=\lVert \psi \rVert ^2{\gt}0\).

Descending induction on \(j\): the outer filters form a descending chain by the induction hypothesis, and replacing \(K_j\) alone keeps the product feasible, so Proposition 21.5.12 applies.

The feasible filters are the continuous image of the product of the compact unitary group and the compact floored simplex, and the uniform filter is feasible. The norm is continuous in the filters, so it attains its maximum on the compact product of the feasible sets. Feasible filters with positive floors are positive definite, hence invertible, and so is their product.

In cut coordinates for \(B\mid B^c\) the operator \(X\) is a sum of at most \(d_S^2\) products \(c\otimes d\) with \(\lVert c\rVert \lVert d\rVert \le c_0\), and Lemma 21.5.11 applies.

The filters on regions disjoint from \(S\) commute with \(X\). The filters on regions containing \(S\) drop out of the expectation by the descending trace argument. If a region splits \(S\), the regions after it contain \(S\) and the regions before it are disjoint from \(S\), so only its conjugation remains, and Lemma 21.5.15 applies; otherwise the expectation is unchanged.

Since \(KHK^{-1}\varphi =E_0\varphi \), the left side is the sum over \(i\) of \(\Re \bigl(\langle \varphi ,h_i\varphi \rangle -\langle \varphi ,Kh_iK^{-1}\varphi \rangle \bigr)\). By Theorem 21.5.13 every filter commutes with the regional state of \(\varphi \) and is clipped relative to it; Lemma 21.5.16 bounds each term, and exchanging the two sums gives the claim.

\(s\lVert \psi \rVert \le \lvert \langle \Omega ,\psi \rangle \rvert =\lvert \langle L_k\Omega ,L_{k-1}\cdots L_0\Omega \rangle \rvert \), and Cauchy–Schwarz applies.

The eigenvalues of \(L\) are nonnegative with \(\sum _j l_j^{2/a}=1\). In cut coordinates \(L\) acts as \(L\otimes 1\), and Lemma 21.5.2 applies.

The optimum is attained by Proposition 21.5.14. By Theorem 21.5.17 and the gap, the normalized output of an optimal prefix of length \(k+1\) has squared overlap at least \(1-E/\Delta \) with \(\Omega \), and Lemma 21.5.18 gives \(\sqrt{1-E/\Delta }\, N_{k+1}\le x_kN_k\). Telescoping from \(N_0=1\) gives the claim.

The filters \(U_j\operatorname {diag}\bigl(((1-\delta )y_j+\delta /d_j)^{a_j/2}\bigr)U_j^*\) are feasible at the floor \(\delta /d_j\) and converge to \(L_j\) as \(\delta \downarrow 0\). The bound of Theorem 21.5.20 does not depend on the floor, and the output norm is continuous in the filters.

On a disjoint union the lift of \(A\otimes B\) is the product of the lifts of \(A\) and of \(B\). The projection \(P_i\) commutes with the filters on the \(T_j\), is idempotent, and does not increase norms.

The function \(F(z)=M_{k-1}^z\cdots M_0^z\xi \) is entire and bounded on the strip \(0\le \Re z\le 1\). On \(\Re z=0\) every factor is unitary. On \(\Re z=1\) the identity \(M_{k-1}^{1+iy}\cdots M_0^{1+iy}=(U_{k-1}\cdots U_0)(V_{k-2}^*M_{k-1}V_{k-2})\cdots M_0\), with \(U_j=M_j^{iy}\) and \(V_j=U_j\cdots U_0\), writes \(F(1+iy)\) as a unitary applied to a nested product of unitary conjugates. The three-lines theorem gives \(\lVert F(x)\rVert \le B^x\). Since \(F(0)=\xi \), the one-sided derivative of \(\lVert F(x)\rVert ^2-B^{2x}\) at \(x=0\) is nonpositive, which is the second claim.

The regional states of \(A\cup B\), \(B\cup C\) and \(B\) are the partial traces of the regional state of \(A\cup B\cup C\), viewed as a tripartite matrix, so strong subadditivity applies.

By Proposition 21.5.21 and Lemma 21.5.19, every nested product of nonnegative filters on the \(D_j\) has output norm at most \(B=\prod _k x_k/\sqrt{1-E/\Delta }^{\, m}\) with \(x_k^2=\exp (-a_kS(D_k)+1024e\vartheta \mathcal B_ka_k^2)\). For a Schmidt index \(i\) with \(p_i{\gt}0\), Lemma 21.5.22 bounds the nested products of filters on the \(T_j\) applied to \(\xi _i=P_i\Omega /\sqrt{p_i}\) by \(B/\sqrt{p_i}\). Theorem 21.5.23 with the filters \(((1-\delta )\rho _{\xi _i,T_j}+\delta /d)^{a_j/2}\) gives, as \(\delta \downarrow 0\), \(-\sum _j\tfrac {a_j}2S_{\xi _i}(T_j)\le \log B-\tfrac 12\log p_i\). Averaging with weights \(p_i\) and using concavity of the entropy gives \(-2\log B\le S(X)+\sum _ja_jS(T_j)\). Hence \(\sum _ja_jS(X\mid T_j)\le S(X)+\text{error}\), and Theorem 21.5.24 bounds the left side below by \(2S(X\mid T)\).

In the power series after its constant term, every exponent is at least two, so \(a^{2k}\le a^2\) for \(k\ge 1\). For positive \(p,q\), set \(z=(\log p-\log q)/2\). The clipping condition and monotonicity of \(\cosh \) in the absolute value of its argument give the bound, using \(\sqrt{pq}\cosh z=(p+q)/2\). If \(a{\lt}0\), the clipping condition forces \(t=0\). If either probability vanishes, the left side vanishes.

Keep the actual number of matrix units on \(S\cap B\) in the construction of Lemma 21.4.1. No enlargement to the dimension of all of \(S\) is needed.

Use the coefficient \(a^2(p_i+p_k)/2\) in the Hermitian pairing. The weighted row and column estimate contributes the factor two. Unitary transport gives the common-basis statement. For supported operators use the matrix-unit decomposition just proved.

Filters outside the crossing region drop out by the descending trace identity; filters inside it have disjoint support and commute with \(X\). Apply the sharp supported bound to the remaining filter. If there is no crossing, the expectation difference is zero.

Expand the expectation difference as a sum over \(i\), bound each summand by its absolute value, and apply Theorem 21.5.1.4. Interchange the finite sums to group the interactions by their crossing region. The eigenvector equation and unit normalization identify the conjugated expectation with \(E_0\).