Properties of the positive square root.

The first three identities are the pointwise identities \(\chi =\bar\chi \), \(\chi ^2=\chi \) and \(\chi (x)\sqrt x=\sqrt x\) under the functional calculus. The identity \(x\chi (x)=x\) gives \(M^\dagger M(\Pi -\mathbb {1})=0\), hence \(M(\Pi -\mathbb {1})=0\). For the ranges, \(\Pi =P\, g(M^\dagger M)\), because \(\sqrt x\, g(x)=\chi (x)\) on the nonnegative spectrum, and \(P=\Pi P\).

\(VP=M\, g(M^\dagger M)P=M\Pi =M\) by Theorem 35.1.1.3. Since \(g(M^\dagger M)\) is Hermitian, \(V^\dagger V=g(M^\dagger M)\, M^\dagger M\, g(M^\dagger M)\), which is the functional calculus of \(x\, g(x)^2=\chi (x)\) on the nonnegative spectrum.

For injective \(M\) the matrix \(M^\dagger M\) is positive definite, so \(\chi =1\) on its spectrum. Then \(V^\dagger V=\Pi =\mathbb {1}\) by Theorem 35.1.1.4, and \(P\) is invertible since \(P^2=M^\dagger M\) is.

Take the polar factors of Definition 35.1.1.1.

If \(K^\dagger KM=M\), then \((KM)^\dagger KM=M^\dagger K^\dagger KM=M^\dagger M\). The positive part and the support projector are functions of \(M^\dagger M\), so they do not change, and the partial isometry of \(KM\) is \(KM\, g(M^\dagger M)=KV\).

Both composites are the identity entrywise.

Let \(\widehat Bx=0\) and consider the linear functional \(\varphi (Y)=\sum _{\alpha ,\beta }Y_{\alpha \beta }x_{\alpha \beta }\) on \(M_{D}(\mathbb {C})\). Since \(\varphi \) vanishes on every \(B^i\), it vanishes on their span \(M_{D}(\mathbb {C})\); evaluating on matrix units yields \(x=0\).

Both sides have \((i,(\alpha ,\beta ))\) entry \(\sum _jW_{ij}B^j_{\alpha \beta }\).

Expand \((B_W)^i=\sum _jW_{ij}B^j\) in \(\sum _i(B_W)^iX((B_W)^i)^\dagger \) and collect \(\sum _i\overline{W_{il}}W_{ij}=(W^\dagger W)_{lj}\). The case \(W^\dagger W=\mathbb {1}\) is the unitary freedom of Kraus representations; the weaker hypothesis \(B_{W^\dagger W}=B\) covers partial isometries.

By definition.

By Lemma 35.1.2.4, the physical matrix of \(P_V\) is \(VP=\widehat B\) by Theorem 35.1.1.4, and a tensor is determined by its physical matrix. The remaining identities are Theorems 35.1.1.2, 35.1.1.3 and 35.1.1.4, reindexed along the identification of \(\{ 0,\dots ,D^2-1\} \) with the virtual pairs; the last one is \(\Pi P=P\) read through Lemma 35.1.2.4.

Write \(B=P_V\) by Theorem 35.1.3.3. Since \(V^\dagger V=\Pi \) and \(P_\Pi =P\), Theorem 35.1.2.5 gives \(\mathcal{E}_{P_V}=\mathcal{E}_P\).

The physical matrix of \(X_W\) is \(W\widehat X\) (Lemma 35.1.2.4), and \(X_{W^\dagger W}=X\) reads \(W^\dagger W\widehat X=\widehat X\). Apply Theorem 35.1.1.7 and reindex. A relabelling by a bijection is the rotation by its permutation matrix, which is an isometry; this gives \(P\) and \(\Pi \). For \(V\) the relabelling is read off directly: relabelling the rows of \(\widehat X\) leaves \(\widehat X^\dagger \widehat X\) unchanged, so it relabels the rows of \(V=\widehat X\, g(\widehat X^\dagger \widehat X)\) (Definition 35.1.1.1).

By Theorem 35.1.2.3, \(\widehat B\) is injective; apply Theorem 35.1.1.5 and reindex.

Since \(M\Pi _S=M\), the matrix \(Q=P(\mathbb {1}-\Pi _S)\) has \(Q^\dagger Q=(\mathbb {1}-\Pi _S)M^\dagger M(\mathbb {1}-\Pi _S)=0\), so \(Q=0\). As \(\Pi =PR\) for some \(R\) by Theorem 35.1.1.3, and \(\Pi \) and \(P\) are Hermitian, \(\Pi (\mathbb {1}-\Pi _S)=R^\dagger Q=0\), that is, \(\Pi =\Pi \Pi _S\). Conversely, \(M\Pi _S(\mathbb {1}-\Pi )=M(\mathbb {1}-\Pi )=0\), and every column of \(\Pi _S(\mathbb {1}-\Pi )\) is supported on \(S\), so \(\Pi _S(\mathbb {1}-\Pi )=0\). Hence \(\Pi =\Pi ^\dagger =\Pi _S\Pi =\Pi _S\).

If \(B\) spans all matrix units, its span is the full matrix algebra, and conversely. For \(x\) supported on \(S\) with \(\widehat Bx=0\), the functional \(Y\mapsto \sum _{\alpha ,\beta }Y_{\alpha \beta }x_{\alpha \beta }\) vanishes on every \(B^i\), hence on every matrix unit in \(S\), so \(x=0\) on \(S\). Then Theorem 35.1.3.7 gives \(\Pi =\Pi _S\), the column identity is \(V^\dagger V=\Pi _S\), and \(P^x=\sum _y(\Pi _S)_{xy}P^y=0\) for \(x\notin S\) because \(\Pi P=P\). Finally \(V=\widehat BR\) for some \(R\) by Lemma 35.17.11; reindexing gives \(G\).

\(\mathcal{E}_P=\mathcal{E}_{A^{[q]}}\) by Theorem 35.1.3.4, and \(\mathcal{E}_{A^{[q]}}=\mathcal{E}_A^q\) by Theorem D.4.8 [ 219 , eq. (6) ] .

By Lemma G.1.1.2, the right-hand side is \(V^{(M)}(P_V)_\sigma \), and \(P_V=A^{[q]}\) by Theorem 35.1.3.3. Finally \(V^{(M)}(A^{[q]})_\sigma =V^{(qM)}(A)_{\widetilde\sigma }\) by Lemma 2.7.2.

The entry at \(((l,r),(\alpha ,\beta ))\) is \((\sqrt\sigma \, |l\rangle \! \langle r|)_{\alpha \beta }=(\sqrt\sigma )_{\alpha l}\, \delta _{\beta r}\), the entry of \((\sqrt\sigma )^T\otimes \mathbb {1}\). The transpose of the positive semidefinite matrix \(\sqrt\sigma \) is positive semidefinite, and so is its Kronecker product with \(\mathbb {1}\).

Write \(S=\sqrt\sigma \), so \(S^\dagger =S\) and \(S^2=\sigma \). Since \(|l\rangle \! \langle r|X|r\rangle \! \langle l|=X_{rr}|l\rangle \! \langle l|\),

Expand the trace of the product \(\prod _k\sqrt\sigma \, |l_k\rangle \! \langle r_k|\) as a sum over cyclic index paths \((t_k)_k\), one index per bond. The entry \((\sqrt\sigma \, |l_k\rangle \! \langle r_k|)_{t_k t_{k+1}}\) vanishes unless \(t_{k+1}=r_k\), so only the path \(t_{k+1}=r_k\) survives, and its product is \(\prod _k(\sqrt\sigma )_{r_{k-1}l_k}\). Reindexing the product along the cyclic shift of the sites gives (2).

By Theorem 35.2.4, \(\mathcal{E}_{P_\infty }\) is the projection \(X\mapsto \operatorname{tr}(X)\, \sigma \), which is idempotent because \(\operatorname{tr}\sigma =1\). The physical blocking isometry follows from Theorem 17.1.6.

With \(S=\sqrt\sigma \), \(\langle \omega | \omega \rangle =\sum _{r,l}\overline{S_{rl}}S_{rl}=\operatorname{tr}(S^\dagger S)=\operatorname{tr}\sigma \).

Expanding \(\langle \omega | \omega ' \rangle ^N\) as a sum over bond configurations \((p_k)_k\) of \(\prod _k\overline{\omega _{p_k}}\omega '_{p_k}\) and reindexing along the bond regrouping of Definition 35.2.8 gives \(\sum _c\overline{\Omega (c)}\Omega '(c)\). The last statement follows from Theorem 35.2.7.

Write \(P=\mathcal{E}_{P_\infty }\), the projection \(X\mapsto \operatorname{tr}(X)\, \sigma \) by Theorem 35.2.4. The powers of an operator of spectral radius less than one tend to zero, so \((\mathcal{E}_A-P)^q(X)\to 0\). Since \(\mathcal{E}_A\) is trace preserving and fixes \(\sigma \), \(\mathcal{E}_A^q=P+(\mathcal{E}_A-P)^q\) for \(q\ge 1\). Finally \(\mathcal{E}_{A^{[q]}}=\mathcal{E}_A^q\) for every tensor by Theorem D.4.8.

The transfer map of a left-canonical tensor is a channel, and it is irreducible because \(A\) is. By the complementary transfer-map gap for primitive maps  [ 216 , “Complementary transfer-map gap for primitive maps” ] , \(\mathcal{E}_A\) minus the projection \(X\mapsto \operatorname{tr}(X)\, \sigma \) has spectral radius less than one. That projection is \(\mathcal{E}_{P_\infty }\) by Theorem 35.2.4, and Theorem 35.2.10 applies.

The matrix of \(\mathcal B_q^\dagger \mathcal B_q\) is a rearrangement of \(\mathcal{E}_A^{\, q}\): \((\mathcal B_q^\dagger \mathcal B_q)_{(\alpha ,\beta ),(\alpha ',\beta ')} =\mathcal{E}_A^{\, q}(|\beta '\rangle \! \langle \beta |)_{\alpha '\alpha }\). By normality and the gauge (3), \(\mathcal{E}_A^{\, q}(X)\to \operatorname{Tr}(X)\rho \), so the rearranged matrices converge to \(\rho ^{\mathsf T}\otimes \mathbb {1}\). The square root of \(\rho ^{\mathsf T}\otimes \mathbb {1}\) is \((\sqrt\rho )^{\mathsf T}\otimes \mathbb {1}\), since the latter is positive semidefinite and squares to the former, and the square root is continuous on positive semidefinite matrices.

The element \(\sqrt c\, \mathbb {1}\) is positive and its square is \(c\, \mathbb {1}\le b\), so operator monotonicity of the square root gives \(\sqrt c\, \mathbb {1}=\sqrt{c\, \mathbb {1}}\le \sqrt b\).

For every real \(t\), \((\sqrt b+t)^2=b+2t\sqrt b+t^2\). By Theorem 35.3.2, \(\sqrt b\ge \sqrt c\), and \(-\varepsilon \le a-b\le \varepsilon \). For the first bound, \((\sqrt b+s)^2\ge b+2s\sqrt c+s^2\ge b+\varepsilon \ge a\). For the second, \(\sqrt b-s\ge \sqrt c-s\ge 0\) and \((\sqrt b-s)^2\le b-2s\sqrt c+s^2\le a+\varepsilon +s^2-2s\sqrt c\le a\). Operator monotonicity of the square root gives \(\sqrt a\le \sqrt b+s\) and \(\sqrt b-s\le \sqrt a\).

Write \(\varepsilon =\lVert a-b\rVert \) and \(s=\sqrt\varepsilon \). The first bound of Lemma 35.3.3 with \(c=0\) gives \(\sqrt a-\sqrt b\le s\), since \(\varepsilon =s^2\). Exchanging \(a\) and \(b\) gives \(\sqrt b-\sqrt a\le s\). Since \(\sqrt a-\sqrt b\) is self-adjoint and lies between \(-s\) and \(s\), its norm is at most \(s\).

Write \(\varepsilon =\lVert a-b\rVert \). If \(\varepsilon {\gt}c\), the bound \(\lVert \sqrt a-\sqrt b\rVert \le \sqrt\varepsilon \) of Theorem 35.3.4 suffices, since \(\sqrt\varepsilon \le \varepsilon /\sqrt c\). Otherwise let \(s=\varepsilon /\sqrt c\le \sqrt c\), so that \(2s\sqrt c=2\varepsilon \) and \(s^2\le \varepsilon \). Both hypotheses of Lemma 35.3.3 hold, so \(\sqrt b-s\le \sqrt a\le \sqrt b+s\), that is, \(\lVert \sqrt a-\sqrt b\rVert \le s\).

If the algebra is zero, take \(K=0\). Otherwise the spectrum of \(b\) is a compact subset of \((0,\infty )\), so \(b\ge c\, \mathbb {1}\) for some real \(c{\gt}0\), and Theorem 35.3.5 gives the bound with \(K=1/\sqrt c\).

Let \(a{\lt}1\) with \(a\le \lvert \lambda _2\rvert \). If \(0{\lt}\lvert \lambda _2\rvert {\lt}1\), then \(r=\lvert \lambda _2\rvert ^{\gamma }{\gt}\lvert \lambda _2\rvert \ge a\), because \(\gamma {\lt}1\). Otherwise \(\lambda _2=0\) and \(r=1\), or \(\lvert \lambda _2\rvert \ge 1\) and \(r=\lvert \lambda _2\rvert ^{\gamma }\ge 1\); in both cases \(r\ge 1{\gt}a\).

Write \(\mathcal{E}_\infty (X)=\operatorname{Tr}(X)\rho \). The eigenvalues \(z\) of \(\mathcal{E}_A-\mathcal{E}_\infty \) other than \(0\) are eigenvalues of \(\mathcal{E}_A\) different from \(1\), so \(\lvert z\rvert \le \lvert \lambda _2\rvert \), and \(\lvert z\rvert {\lt}1\) by normality. Taking \(a=\lvert z\rvert \) gives \(\lvert z\rvert {\lt}r\). Hence the spectral radius of \(\mathcal{E}_A-\mathcal{E}_\infty \) is less than \(r\), and \(\mathcal{E}_A^{\, n}-\mathcal{E}_\infty =(\mathcal{E}_A-\mathcal{E}_\infty )^n\) gives the first bound for \(n\ge 1\). For \(n=0\) it reads \(\lVert X-\mathcal{E}_\infty (X)\rVert \le C\lVert X\rVert \), which holds once \(C\ge 1+\lVert \mathcal{E}_\infty \rVert \); enlarge \(C\) accordingly.

The transfer map of the \(q\)-site blocked tensor is \(\mathcal{E}_A^{\, q}\), so the entries of \(\mathcal B_q^\dagger \mathcal B_q\) are entries of \(\mathcal{E}_A^{\, q}\) applied to matrix units: \((\mathcal B_q^\dagger \mathcal B_q)_{(\alpha ,\beta ),(\alpha ',\beta ')} =\mathcal{E}_A^{\, q}(|\beta '\rangle \! \langle \beta |)_{\alpha '\alpha }\). In the same way \((\rho ^{\mathsf T}\otimes \mathbb {1})_{(\alpha ,\beta ),(\alpha ',\beta ')} =\operatorname{Tr}(|\beta '\rangle \! \langle \beta |)\, \rho _{\alpha '\alpha }\). Hence every entry of \(\mathcal B_q^\dagger \mathcal B_q-\rho ^{\mathsf T}\otimes \mathbb {1}\) is an entry of \(\mathcal{E}_A^{\, q}(X)-\operatorname{Tr}(X)\rho \) at a matrix unit \(X\), and is at most a fixed multiple of \(Cr^q\) by the first bound. Bounding the operator norm by the sum of the moduli of the finitely many entries gives \(\lVert P_q^2-P_\infty ^2\rVert =\lVert \mathcal B_q^\dagger \mathcal B_q-\rho ^{\mathsf T}\otimes \mathbb {1}\rVert \le Kr^q\). Since \(P_\infty ^2=\rho ^{\mathsf T}\otimes \mathbb {1}\) is positive definite, Theorem 35.3.6 gives \(L\ge 0\) with \(\lVert P_q-P_\infty \rVert \le LKr^q\), so \(K'=LK\).

Since \(V\) is an isometry, its columns \(V|\gamma ,\delta \rangle \) are orthonormal. Place them at the positions \(\iota (\gamma ,\delta )\) and extend them to an orthonormal basis of \((\mathbb {C}^d)^{\otimes q}\); the matrix with these columns is the unitary \(U\). Expanding \(\bigl(\bigotimes _kU_k\bigr)\sum _c\Omega (c)\bigotimes _k|\iota (c_k)\rangle \) gives \(\sum _c\Omega (c)\bigotimes _kV|c_k\rangle =V^{\otimes M}\bigotimes _k|\omega \rangle _{R_kL_{k+1}}\), which is \(|\widetilde\phi _N\rangle \) by Definition 35.3.8.

The matrix elements of \(V\) are \(\bigl(j^{\mathsf T}(\mathbb {1}_D\otimes A_1^{i_1})\cdots (\mathbb {1}_D\otimes A_q^{i_q})G\bigr)_x\) with \(j=\sum _\gamma |\gamma \rangle \otimes |\gamma \rangle \). Apply Theorem 34.2.8 to the row matrix of \(j\) and the sites \(\mathbb {1}_D\otimes A_p^i\); this gives isometric sites \(Q_1,\dots ,Q_q\) and a remainder \(C=RG\) with \(V=Q\, C\) in the sense of (2). By (3), \(V^\dagger V=C^\dagger C\), so \(C\) is an isometry because \(V\) is. Absorb \(C\) into the last site, \(X_q^i=Q_q^iC\) and \(X_p=Q_p\) for \(p{\lt}q\); the last site is again isometric, \(\sum _i C^\dagger (Q_q^i)^\dagger Q_q^iC=C^\dagger C=\mathbb {1}\).

Expand the two matrix products in (6) and interchange the finite sums over \(\beta ,\gamma ,\delta \). Transposition reverses the order of matrix multiplication.

Apply successive decompositions to the initial row \(\sum _\alpha \langle \alpha ,\alpha |\) and the matrices \(\mathbb {1}_D\otimes A_p^i\). Restrict the remainder to its nonzero rows. Preservation of inner products along the isometric sites gives \(E^\dagger E=\mathbb {1}\).

Independent inward decompositions of the left and right virtual-pair coefficient matrices give \(L=E_L T_L\) and \(R=E_R T_R\). Set \(C=(T_L\otimes \mathbb {1}_d\otimes T_R)F\). Matrix multiplication gives (7). Writing \(E=E_L\otimes \mathbb {1}_d\otimes E_R\), the identity \(C^\dagger C=C^\dagger E^\dagger EC=V^\dagger V=\mathbb {1}\) proves that the central map is an isometry.

By Theorem 35.1.3.6, \(V^\dagger V=\mathbb {1}\). Lemma 35.21.1.6 expresses \(V=BP^{-1}\) in the matrix-product coefficients of the given chain. Split the chain at the chosen site and orient its right part inward. Equation (6), with \(G=P^{-1}\), gives these same coefficients. Apply the two-sided factorization theorem.

Interchange finite sums to commute the input change with the central contraction. The existing polar factorization gives \(V=BG\) for a virtual-pair matrix \(G\), and splitting the chain at the chosen site gives exactly the central product. Reindexing physical configurations preserves column inner products.

Direct multiplication gives both normalization identities. The products \(B^1B^2,B^0B^1,B^0B^2,B^2B^1\) are respectively one half of the four matrix units, proving two-site injectivity. Since \(B^3=0\) and \(V=BG\), the final physical row of \(V\) is zero. If the square matrix \(V\) were an isometry, it would also satisfy \(VV^\dagger =\mathbb {1}_4\), contradicting its final diagonal entry.

Parametrize the range of the Hermitian idempotent \(\Pi \) by orthonormal coordinates \(J\), so \(J^\dagger J=\mathbb {1}_s\) and \(JJ^\dagger =\Pi \). Then \((VJ)^\dagger (VJ)=J^\dagger \Pi J=\mathbb {1}_s\). Apply the existing two inward sweeps to the central factor \(GJ\). Their isometric evaluations force the central map to be an isometry on the support input. Finally \(V\Pi =V\) gives the reconstruction identity in (8).

Since \(\mathcal B_q\) is injective, \(P\) is positive definite and \(V=\mathcal B_qP^{-1}\), so \(\langle i_1\cdots i_q|V|x\rangle =\sum _{\alpha ,\beta }(A^{i_1}\cdots A^{i_q})_{\alpha \beta }(P^{-1})_{(\alpha ,\beta ),x}\). This is an open-boundary matrix product map with bond dimension at most \(D^2\): the site matrices are \(\mathbb {1}_D\otimes A^i\) (the source writes \(A^i\otimes \mathbb {1}_D\); the order of the two bond factors only permutes the bond index), the two bond factors are joined at the left end, and \(P^{-1}\) is absorbed into the right end [ 219 , eq. (13) ] . Theorem 35.4.1 with \(G=P^{-1}\) makes every site isometric by successive decompositions from the left [ 219 , eq. (14) ] ; the remainder at the input end is an isometry because \(V\) is [ 219 , eq. (15) ] . The sites of that theorem are numbered from the output end and those here from the input end, so its \(p\)-th site is the site \(q+1-p\) here, \(X_p^i=W_{q+1-p}^i\) for \(p=1,\dots ,q\), which turns \((X_1^{i_1}\cdots X_q^{i_q})_{1x}\) into \((W_q^{i_1}\cdots W_1^{i_q})_{1x}\) and \(b_p\) into \(D'_{q+1-p}\).

Since \(EQ=Q\) and \(WE=W\), \(M^\dagger M=QE\, Q=Q^2\), so \(Q\) is the positive square root of \(M^\dagger M\). Two orthogonal projectors with the same range coincide, which identifies \(E\). Finally \((V-W)Q=M-M=0\) for the partial isometry \(V\) of \(M\), and both \(V\) and \(W\) are fixed by \(E\), whose range lies in the range of \(Q\); hence \(V=W\).

Blocking commutes with maps on the physical leg: blocking \(C_W\) over \(m\) sites gives the tensor \(C\) blocked over \(m\) sites, rotated by \(W^{\otimes m}\) (Theorem G.1.1.3). Blocking \(2^{k}\) sites is blocking \(2^{k-1}\) pairs of the two-site blocked tensor, and the outer layer composes with the maps of the inner tree by \((A_{W'})_W=A_{WW'}\) (Lemma 11.7.3), so induction on \(k\) gives (12). For the polar factors, write the block of \(2^{k}\) sites as the first layer \(K=\bigl(V^{(1)}\bigr)^{\otimes 2^{k-1}}\) applied to the tensor \(T_1\) blocked over \(2^{k-1}\) sites, regrouped along neighbouring pairs. The range of that blocked tensor lies in \((\operatorname {ran}P_1)^{\otimes 2^{k-1}}\), because \(\Pi _1\) fixes the tensor of \(P_1\), so \(K^\dagger K=\Pi _1^{\otimes 2^{k-1}}\) fixes it. By Theorem 35.1.3.5, the isometric factor of the block is \(K\) times the isometric factor of \(T_1\) blocked over \(2^{k-1}\) sites, and the positive parts and the support projectors agree, also after the regrouping. Induction on \(k\) gives the identification with \(V\), \(P_q\) and the support projector of \(\mathcal B_q\), and \(V^\dagger V=\Pi \) gives the initial projector of the product of the layers.

By Theorem 35.3.7, \(\lVert P_q-P_\infty \rVert \le K_2e^{-\gamma q/\xi }\).

Let \(\tau _q\) and \(\tau _\infty \) be the transfer matrices of the mixed transfer maps of \(P_q\) and of \(P_\infty \) against \(P_\infty \). Then \(\langle \phi _M(P_\infty )|\phi _M(P_q)\rangle =\operatorname{Tr}\tau _q^M\), \(1=\operatorname{Tr}\tau _\infty ^M\), \(\tau _\infty ^2=\tau _\infty \), and \(\delta =\lVert \tau _q-\tau _\infty \rVert \le K_3e^{-\gamma q/\xi }\), since \(\tau _q-\tau _\infty \) depends linearly on \(P_q-P_\infty \). Writing \(\tau _q^M-\tau _\infty ^M=\sum _{k{\lt}M}\tau _q^k(\tau _q-\tau _\infty )\tau _\infty ^{M-1-k}\) and bounding \(\lVert \tau _q^k\rVert \le c+\lVert \tau _q^k-\tau _\infty ^k\rVert \) with \(c\ge \max (1,\lVert \tau _\infty \rVert )\), induction on \(M\) gives \(\lVert \tau _q^M-\tau _\infty ^M\rVert \le c\bigl((1+c\delta )^M-1\bigr) \le c\, Mc\delta \, e^{Mc\delta }\). Since \(\langle \phi _M(P_\infty )|\phi _M(P_q)\rangle -1=\operatorname{Tr}\tau _q^M-\operatorname{Tr}\tau _\infty ^M\), this bounds the overlap itself, and hence also its modulus, which proves the explicit bounds. When \(y\le 1\) it is at most \(Ce^{C}y\). When \(y{\gt}1\), the overlap is at most \(\lVert \phi _M(P_q)\rVert =\lVert \phi _N(A)\rVert \), and these norms are bounded because \(\lVert \phi _N(A)\rVert ^2=\operatorname{Tr}\mathcal{E}_A^{\, N}\) converges to \(1\).

If \(x\geq 1\), then \(y\geq 1\) and \(Cy\geq 1\). If \(x{\lt}1\) and \(L_j{\gt}q\), then \(x^{L_j}\leq x^q\leq y\), so \(Cy\geq x^{-L_j}y\geq 1\). When \(Cy\geq 1\) the bound follows from \(\varepsilon \leq 1\leq Cy\leq Cy\, e^{Cy}\).

\(KMe^{-x}=e^{\log K+\log M-x}\leq e^{\log \varepsilon }=\varepsilon \).

Since \(\log q\geq 0\), \(q/a\geq \log M+\log q-\log \varepsilon +\log K\geq \log K+\log M-\log \varepsilon \); apply Lemma 35.5.3 with \(x=q/a\).

Put \(T=e+Z\), \(W=fZf\), \(\beta =\lVert fZe\rVert \), \(\beta '=\lVert eZf\rVert \), \(\zeta =\lVert W\rVert \), and \(\rho =1+\beta \beta '/(1-\zeta )\). For \(P\in \{ e,f\} \) the blocks \(x_M=PT^Me\) and \(y_M=PT^Mf\) satisfy \(x_{M+1}=x_M+y_M\, fZe\) and \(y_{M+1}=x_M\, eZf+y_MW\), because \(eTe=e\). Induction on \(M\) gives \(\lVert eT^Me-e\rVert \le \lVert e\rVert (\rho ^M-1)\) and \(\lVert eT^Mf\rVert \le \lVert e\rVert \beta '\rho ^M/(1-\zeta )\), and also \(\lVert fT^Me\rVert \le \lVert f\rVert \beta (\rho ^M-\zeta ^M)/(1-\zeta )\) and \(\lVert fT^Mf-fW^M\rVert \le \lVert f\rVert \beta \beta '(\rho ^M-\zeta ^M)/(1-\zeta )^2\). Since \(\operatorname{Tr}x=\operatorname{Tr}(exe)+\operatorname{Tr}(fxf)\), it follows that \(\operatorname{Tr}T^M-\operatorname{Tr}e\) is the sum of the traces of \(eT^Me-e\), of \(fT^Mf-fW^M\), and of \(fW^M\), the last of norm at most \(\lVert f\rVert \zeta ^M\le \lVert f\rVert z^2\). With \(z\le 1/2\) one has \(\rho \le 1+2z^2\) and \(\beta \beta '/(1-\zeta )^2\le 4z^2\), and \((1+2z^2)^M-1\le 2Mz^2e^{2Mz^2}\).

By Theorem 35.3.7, every linear function of the tensor \(P_q\) is within \(K'e^{-\gamma q/\xi }\) of its value at \(P_\infty \). Let \(\tau _q\) and \(\tau _\infty \) be as in the proof of Lemma 35.5.1, so that the overlap is \(\operatorname{Tr}\tau _q^M\). The idempotent \(\tau _\infty \) is the transfer matrix of the rank-one map \(X\mapsto \operatorname{Tr}(X)\rho \); hence \(\tau _\infty \tau _q\tau _\infty =\alpha \tau _\infty \) with \(\alpha =\operatorname{Tr}\sum _iP_q^i\rho P_\infty ^{i\dagger }=\langle v_\infty | v_q \rangle \), where \(v_q\) and \(v_\infty \) are the vectors \((P^i\sqrt\rho )_i\). Both are unit vectors, because \(\mathcal{E}_{P_q}=\mathcal{E}_A^q\) fixes \(\rho \) and \(\operatorname{Tr}\rho =1\). Hence \(1-\lvert \alpha \rvert \le 1-\operatorname {Re}\alpha =\tfrac 12\lVert v_q-v_\infty \rVert ^2=O(e^{-2\gamma q/\xi })\), and \(\lvert 1-\alpha \rvert \le \lVert v_q-v_\infty \rVert \). Write \(\tau _q=\alpha (\tau _\infty +Z)\). Then \(\tau _\infty Z\tau _\infty =0\) and \(\lVert Z\rVert =O(e^{-\gamma q/\xi })\), so Theorem 35.5.5 bounds \(\lvert \operatorname{Tr}(\tau _\infty +Z)^M-1\rvert \) by a multiple of \(y\, e^{c y}\), and \(\lvert \alpha \rvert ^M\ge 1-M(1-\lvert \alpha \rvert )\) is at least \(1\) minus a multiple of \(y\). For \(M=1\), the vector \(\phi _1(P_\infty )\) is a unit vector, and \(\lVert u\rVert ^2-\lvert \langle w | u \rangle \rvert ^2\le \lVert u-w\rVert ^2\) for every unit vector \(w\); here \(\lVert \phi _1(P_q)-\phi _1(P_\infty )\rVert =O(e^{-\gamma q/\xi })\), since \(\phi _1\) is linear in the tensor.

Write \(c_N=\lVert \phi _N(A)\rVert \). For \(q\) at least the injectivity length of \(A\), the blocked tensor is injective, \(V\) is an isometry, and \(|\widetilde\phi _N\rangle =V^{\otimes M}|\phi _M(P_\infty )\rangle \) is normalized. Since \(|\phi _N(A)\rangle =V^{\otimes M}|\phi _M(P_q)\rangle \) by Theorem 35.1.3.11, this gives \(c_N\langle \widetilde\phi _N | \phi _N \rangle =\langle \phi _M(P_\infty )|\phi _M(P_q)\rangle =:a\), and \(c_N=\lVert \phi _M(P_q)\rVert \). With \(t=\lvert a\rvert /c_N\le 1\),

which is at most \(2(c_N^2-\lvert a\rvert ^2)\) once \(c_N^2\ge 1/2\). The normalization satisfies \(\lvert c_N^2-1\rvert =\lvert \operatorname{Tr}\mathcal{E}_A^{\, N}-1\rvert \le Ke^{-\gamma ' N/\xi }\) for every \(0{\lt}\gamma '{\lt}1\); we use it at \(\gamma '=\gamma \). For \(M=1\), Theorem 35.5.6 bounds \(c_N^2-\lvert a\rvert ^2\) directly. For \(M\ge 2\), \(c_N^2-\lvert a\rvert ^2\le \lvert c_N^2-1\rvert +2\max (0,1-\lvert a\rvert )\), with \(\lvert c_N^2-1\rvert \le Ke^{-2\gamma q/\xi }\) because \(N\ge 2q\), and \(1-\lvert a\rvert \) is bounded by Theorem 35.5.6. The finitely many \(q\) below the injectivity length, and the case \(Cy\ge 1\), are covered by \(\varepsilon \le 1\).

Since \(\gamma /2{\lt}1\), apply the second-order estimate at rate \(\gamma /2\) to obtain \(\varepsilon \leq C_\gamma M e^{-\gamma q/\xi }\). The ceiling condition gives \(\gamma q\geq \xi (1+\eta )\log N\). Since \(q\geq 1\) and \(M\leq N\), \(C_\gamma M e^{-\gamma q/\xi }\leq C_\gamma N e^{-(1+\eta )\log N} =C_\gamma N^{-\eta }\).

Apply the second-order approximation estimate at the fixed rate \(\gamma =1/4\), obtaining \(\varepsilon \leq C M e^{-q/(2\xi )}\). Formula (16) gives \(q\geq 2\xi (1+\eta )\log N\). Since \(N=Mq\geq 2\), one has \(q\geq 1\) and \(M\leq N\). Thus

This proves (17). The second-order rate is the justification for the exact coefficient in (16).

An ordered product over the sites of the ring is the ordered product over the blocks of the products over their sites, so \(\phi _N(A)(s)=\operatorname{Tr}(\mathcal B_{\ell _1}^{s_1}\cdots \mathcal B_{\ell _M}^{s_M})\), and \(\mathcal B_{\ell _k}=V_{\ell _k}P_{\ell _k}\) gives \(|\phi _N(A)\rangle =(\bigotimes _kV_{\ell _k})|\phi _M(P_{\ell _1},\dots ,P_{\ell _M})\rangle \), the periodic state with the tensor \(P_{\ell _k}\) at site \(k\). Isometries \(W_k\), one on each site, preserve inner products: \(\sum _s\overline{((\bigotimes _kW_k)\psi )(s)}((\bigotimes _kW_k)\varphi )(s) =\langle \psi | \varphi \rangle \), since \(\sum _{s_k}\overline{(W_k)_{s_ka}}(W_k)_{s_kb}=\delta _{ab}\) factor by factor. This gives the norm by Theorem 35.2.9, and reduces the overlap to \(\sum _u\operatorname{Tr}(P_{\ell _1}^{u_1}\cdots P_{\ell _M}^{u_M}) \overline{\operatorname{Tr}(P_\infty ^{u_1}\cdots P_\infty ^{u_M})}\) by Theorem 35.2.5. Expanding the trace of \(\tau _{\ell _1}\cdots \tau _{\ell _M}\) as a sum over cyclic paths of pairs of bond indices, with \((\tau _\ell )_{(j,i),(l,m)}=\sum _a(P_\ell ^a)_{im}\overline{(P_\infty ^a)_{jl}}\), and summing the two paths separately gives the same sum, as in Theorem 18.6.2.

Expand \(\lVert x-y\rVert ^2=\lVert x\rVert ^2+\lVert y\rVert ^2 -2\operatorname {Re}\langle x,y\rangle \) and use the unit norms.

Write \(G\) for the positive polar matrix of the physical matrix of \(B\). The trace identity in Lemma 35.17.5.2 gives \(\alpha =\operatorname{Tr}(G((\sqrt\sigma )^{\mathsf T}\otimes \sigma ))\). Both factors are positive semidefinite, so their trace product is real and nonnegative.

The positive polar tensor has the same transfer map as \(B\). Thus \(\lVert \iota (P)\rVert =\lVert \iota (P_\infty )\rVert =1\) and \(\langle \iota (P_\infty ),\iota (P)\rangle =\alpha \). Apply Theorem 35.5.1.1; the nonnegativity of the squared distance also gives \(\operatorname {Re}\alpha \le 1\).

Put \(P_m=\prod _{j=0}^{m-1}(e+Z_j)\) and \(W_j=fZ_jf\). For \(a=e,f\), the blocks \(x_m=aP_me\) and \(y_m=aP_mf\) satisfy \(x_{m+1}=x_m+y_mfZ_me\) and \(y_{m+1}=x_meZ_mf+y_mW_m\). With \(\rho =1+z^2/(1-z)\), induction gives \(\lVert eP_me-e\rVert \le \lVert e\rVert (\rho ^m-1)\) and \(\lVert eP_mf\rVert \le \lVert e\rVert z\rho ^m/(1-z)\). The analogous recurrences starting from \(f\) give

Cyclicity gives \(\tau (P_M)=\tau (eP_Me)+\tau (fP_Mf)\). Since \(M\ge 2\), the complementary product has norm at most \(\lVert f\rVert z^M\le \lVert f\rVert z^2\). Now use \(\rho \le 1+2z^2\) and \((1+2z^2)^M-1\le 2Mz^2e^{2Mz^2}\).

Choose \(C\ge 4\) large enough in terms of \(K\), \(\lVert e\rVert \) and \(\lVert 1-e\rVert \). The smallness condition gives \(\delta \le 1\) and \(|\alpha _j|\ge 3/4\). Set \(Z_j=\alpha _j^{-1}T_j-e\). Then \(eZ_je=0\) and \(\lVert Z_j\rVert \le 2(1+\lVert e\rVert )\delta \). Its three complementary blocks are bounded by a fixed multiple of \(\delta \), so Theorem 35.5.1.4 applies. Put \(a=\prod _{j=0}^{M-1}\alpha _j\) and \(P=\prod _{j=0}^{M-1}(e+Z_j)\). Then \(|a|\le 1\), \(|a-1|\le M\delta ^2e^{M\delta ^2}\), and \(\tau (\prod _{j=0}^{M-1}T_j)-1=a(\tau (P)-1)+(a-1)\). Enlarging \(C\) gives the assertion.

Induct on \(M\), separating the last factor in the ordered product and the last summand.

Every power of \(e\) has norm at most \(c\). Apply (22) and induct on \(M\), using \(\lVert \prod _{i=0}^{j-1}X_i\rVert \le c+c((1+c\delta )^j-1)\) in each summand.

For \(v=0\), the second assumption gives \(b\ge 1\). Otherwise put \(c=\lVert v\rVert {\gt}0\) and \(w=|\langle \psi ,\widehat v\rangle |\le 1\). Since \(cw=|\langle \psi ,v\rangle |\), \(1-w=1-|\langle \psi ,v\rangle |+(c-1)w\). Use \((c-1)w\le |c-1|\le |c^2-1|\).

Each mixed transfer matrix is within \(Ke^{-\gamma \ell /\xi }\) of the fixed-point idempotent \(e\), whose positive powers have trace one. Since \(\ell _j\ge q\), apply Theorem 35.5.1.7 with \(\delta =Ke^{-\gamma q/\xi }\) and then the boundedness of the trace.

The fixed-point transfer matrix \(e\) satisfies \(e^2=e\) and \(\operatorname{Tr}e=1\). Each factor has scalar compression \(e\tau _\ell e=\alpha _\ell e\). Theorems 35.5.1.2 and 35.5.1.3 give \(0\le \alpha _\ell \le 1\) and \(1-\alpha _\ell =\lVert \iota (P_\ell )-\iota (P_\infty )\rVert ^2/2\). Both the mixed-transfer difference and the weighted tensor difference are bounded by \(Ke^{-\gamma q/\xi }\), uniformly in \(\ell \ge q\). Apply Theorem 35.5.1.5 and absorb \(K^2\) into \(C\).

Put \(v=\phi _N(A)\), \(c_N=\lVert v\rVert \) and \(a=\langle \psi _{\ell ,\omega },v\rangle \). The block isometries give \(\lVert \psi _{\ell ,\omega }\rVert =1\) and \(a=\operatorname{Tr}(\tau _{\ell _0}\cdots \tau _{\ell _{M-1}})\). As in Theorem 35.5.7, once \(c_N^2\ge 1/2\), normalization gives \(\varepsilon \le 2(c_N^2-|a|^2)\). Write \(u=Me^{-2\gamma q/\xi }\). If \(M\ge 2\), Theorem 35.5.1.10 bounds \(1-|a|\) by \(Cu e^{Cu}\) in the small-error regime. The transfer gap gives \(|c_N^2-1|\le K e^{-\gamma N/\xi }\); since \(N\ge Mq\ge 2q\), this is at most \(K e^{-2\gamma q/\xi }\). If \(M=1\), Theorem 35.5.6 bounds \(c_N^2-|a|^2\) directly by \(K e^{-2\gamma N/\xi }\), hence by \(K e^{-2\gamma q/\xi }\). The first-order normalization estimate also yields \(c_N^2\ge 1/2\) when a sufficiently large fixed multiple of \(u\) is below one. The remaining regime is covered by \(\varepsilon \le 1\). Finally \(Cu e^{Cu}\le Ce^C u\) for \(u\le 1\), giving the assertion after increasing the constant.

The case \(D=0\) is trivial. If \(d\ge 2\) then \(D{\lt}d^D\). If \(d\le 1\) and \(D\ge 1\), injectivity of the blocked tensor gives \(D^2\le d^q\), which forces \(d=D=1\). In either case \(D\le d^D\), so each bond index can be encoded in \(D\) sites. Encode the two bond indices of a block in its first and last \(D\) sites, and write \(|l,0,r\rangle \) for the block with \(l\) on the first \(D\) sites, \(r\) on the last \(D\) sites, and \(|0\rangle \) elsewhere. This defines an injective map \(E\) from bond-pair configurations to physical configurations. Its restriction to block \(k\) is the prescribed \(|l_k,0,r_k\rangle \); its restriction to the pair window is \((r_k,l_{k+1})\). These pair-window restrictions, together with the zero values off the windows, characterize \(E\). Within each layer the windows are disjoint, so their embedded operators commute and the layer’s matrix coefficients factor over the windows. The all-zero product input has coefficient \(\prod _i\delta _{s_i,0}\). For equal block length \(q\), the site \((k,j)\) is \(kq+j\), and reading a configuration block by block agrees with the uniform blocking equivalence. A first layer of unitaries, one on the last \(D\) sites of each block and the first \(D\) sites of the next, sends \(|0\cdots 0\rangle \) to \(\bigotimes _k|\omega \rangle _{R_kL_{k+1}}|0\rangle _{C_k}\) [ 219 , eq. (12) ] ; by Theorem 33.2.1 it has constant depth. On each block \(k\), in parallel, a unitary \(U_k\) with \(U_k|l,0,r\rangle =V_{\ell _k}|l,r\rangle \) is applied, as in Theorem 35.3.9. SWAP gates move the first \(D\) sites next to the last ones in \(O(q)\) gates. By Theorem 35.4.12, \(V_\ell \) is the ordered product of \(\ell \) isometries, each from the bond space into the bond space and one site; each is extended to a unitary on the bond register of \(2D\) sites and one fresh site, and the register moves one site to the left at each step, so the staircase consists of \(q-2D\) unitaries on \(2D+1\) sites followed by one unitary on the first \(2D\) sites [ 219 , paragraph “The sequential-RG circuit” ] . Each of them has bounded depth by Theorem 33.2.1, so \(U_k\) has depth \(O(\ell _k)\le O(L)\), and circuits on disjoint blocks run in the same layers. A configuration contributes to the pair layer applied to \(|0\cdots 0\rangle \) only if it carries a pair on every window and \(|0\rangle \) elsewhere, which identifies the output with (18).

Write \(P(X)=\operatorname{tr}(X)\sigma \) and \(R=\mathcal{E}_A-P\). Trace preservation and stationarity give \(P^2=P\) and \(PR=RP=0\). Normality and the spectral assumption imply that every eigenvalue of \(R\) is zero. Its generalized zero-eigenspace is therefore the entire \(D^2\)-dimensional matrix space, so \(R^{D^2}=0\). For \(n\geq D^2\), the identity \(\mathcal{E}_A^n=P+R^n\) gives (23).

The physical matrix satisfies

The second matrix is the physical matrix of \(P_\infty (\sigma )\). When \(\sigma {\gt}0\), the Gram matrix in (24) is invertible. Multiplication by it is surjective on the virtual pair space, so every virtual matrix is a linear combination of the matrices \(B^i\).

The local polar decompositions give \(A^{[\ell _k]}=V_kP_\infty (\sigma )\). Contracting these identities around the ring yields \(|\phi _N(A)\rangle =(\bigotimes _kV_k)|\Omega _\sigma \rangle \), precisely the fixed-point pair construction. This is the exact-factor case of the construction in [ 219 ] .

Every block has transfer map \(X\mapsto \operatorname{tr}(X)\sigma \) by (23). Its positive polar tensor is therefore \(P_\infty (\sigma )\). Apply Theorem 35.6.1.3.

Write \(N=Mq_0+r\) with \(M\geq 1\) and \(0\leq r{\lt}q_0\). Choose \(M-1\) blocks of length \(q_0\) and one block of length \(q_0+r\). Thus \(q_0\leq \ell _k{\lt}2q_0\), so all blocks have at least \(D^2\) sites and at least \(3D\) sites. Their transfers are exactly \(X\mapsto \operatorname{tr}(X)\sigma \), which gives their positive polar factors and injectivity. The fixed-point pair has norm one, and the block isometries preserve that norm. Exact reconstruction therefore gives \(\| \phi _N(A)\| =1\). The block preparation theorem gives depth at most \(2C(d,D)q_0\), independent of \(A\), \(\sigma \), and \(N\).

Let \(\mathcal B_N=VP\). Then \(\phi _N(A)(s)=\operatorname{Tr}\mathcal B_N^s =\sum _{l,r}\langle s|V|l,r\rangle \operatorname{Tr}P^{(l,r)}\), so \(|\phi _N\rangle \) is the state (18) of a single block with the pair \(\omega (r,l)=\operatorname{Tr}P^{(l,r)}/\lVert \phi _N(A)\rVert \), whose right leg is joined to its own left leg around the ring; \(\omega \) is a unit vector because \(|\phi _N\rangle \) is, by Lemma 35.5.11. Theorem 35.6.1 with \(M=1\) and \(L=N\) gives the first statement. For the second, a unit vector \(|\psi \rangle \) is the first column of a unitary, which by Theorem 33.2.1 is a product of at most \(K_N\) gates on neighbouring sites; take \(K=\sum _{N\le n_0}K_N\).

Bring \(A\) into the gauge (3) as in the proof of Theorem 35.8.2.14, with fixed point \(\sigma \), a bound \(t{\lt}1\) on the moduli of the eigenvalues other than \(1\), and \(r=-\log t\); the error is unchanged by Lemma 35.8.2.9. Let \(K\) be the constant of Theorem 35.5.1.11 at \(\gamma =1/2\), so the error is at most \(KMe^{-rq}\), and let \(L_0\) be a length at which the blocked tensor is injective. Put \(a=1/r\), \(b=\max (\log K,0)/r+L_0+3D+1\), \(Q=a\log (N/\varepsilon )+b\) and \(q=\lceil Q\rceil \), so \(Q\le q{\lt}2Q\) and \(Q\le (a+b/\log 2)\log (N/\varepsilon )\).

If \(q\le N\), write \(N=(M-1)q+q'\) with \(M=\lfloor N/q\rfloor \) and \(q'=q+(N\bmod q)\), so \(q\le q'{\lt}2q\), and cut the ring into \(M-1\) blocks of length \(q\) followed by one of length \(q'\), as in the Supplemental Material of [ 219 ] , also used in its lower-bound proof of Theorem 1. All blocked tensors are injective since \(q\ge L_0\), and Theorem 35.5.1.11 gives error at most \(KMe^{-rq}\le KNe^{-rQ}=Ke^{-rb}\varepsilon \le \varepsilon \). By Theorem 35.6.1 with \(L=2q\) the depth is at most \(2Cq\le 4CQ\).

If \(N{\lt}q\), then \(N{\lt}Q\). If \(N\ge L_0+3D\), the state \(|\phi _N\rangle \) is prepared exactly in depth at most \(C'N\le C'Q\) by Theorem 35.6.1.6; otherwise \(N\) is bounded by a constant depending only on \(A\), and the same theorem prepares \(|\phi _N\rangle \) exactly in depth at most \(K'\le (K'/\log 2)\log (N/\varepsilon )\). In every case the depth is at most \(c\log (N/\varepsilon )\) with \(c=(4C+C')(a+b/\log 2)+K'/\log 2\). The second statement follows from Lemma 35.6.1.8.

In the gauge (3), which multiplies \(|\phi _N(A)\rangle \) by a nonzero scalar, \(\lVert \phi _N(A)\rVert ^2=\operatorname{Tr}\mathcal{E}_A^{\, N}\) differs from \(1\) by at most \(Ke^{-\gamma N/\xi }\) for \(0{\lt}\gamma {\lt}1\), as in the proof of Theorem 35.5.7, which is less than \(1\) for large \(N\).

For unit vectors \(a\), \(b\) and \(\lvert c\rvert =1\), \(\lVert a-cb\rVert ^2=2-2\operatorname {Re}(c\langle a | b \rangle ) \ge 2\varepsilon (a,b)\), with equality for \(c=\overline{\langle a | b \rangle }/\lvert \langle a | b \rangle \rvert \) (any \(c\) if \(\langle a | b \rangle =0\)). Choose \(c_1\), \(c_2\) attaining equality for \((x,y)\) and \((y,z)\). Then \(\lVert x-c_1c_2z\rVert \le \lVert x-c_1y\rVert +\lVert y-c_2z\rVert \), and \((s+t)^2\le 2(s^2+t^2)\) gives \(2\varepsilon (x,z)\le \lVert x-c_1c_2z\rVert ^2 \le 4\bigl(\varepsilon (x,y)+\varepsilon (y,z)\bigr)\).

By Theorem 35.6.1.7 with error \(\varepsilon /4\) and Lemma 33.1.9, there are local circuits \(U_A\), \(U_B\) and unit vectors \(\psi _A=aU_A|0\cdots 0\rangle \), \(\psi _B=bU_B|0\cdots 0\rangle \) with \(\varepsilon (\psi _A,\hat\phi _N(A))\) and \(\varepsilon (\psi _B,\hat\phi _N(B))\) at most \(\varepsilon /4\), of depths \(T_A+2\) and \(T_B+2\) with \(T_A,T_B\) at most a constant times \(\log (4N/\varepsilon )\le 3\log (N/\varepsilon )\). Put \(U=U_BU_A^\dagger \), a local circuit of depth \(T_A+T_B+4\) by Lemma 33.1.8; the term \(4\) is at most \((4/\log 2)\log (N/\varepsilon )\). Then \(U\psi _A=(a/b)\psi _B\) is a unit vector, and since \(U\) is unitary, \(\varepsilon (U\hat\phi _N(A),(a/b)\psi _B)=\varepsilon (\hat\phi _N(A),\psi _A)\). Lemma 35.6.1.9 gives \(\varepsilon (U\hat\phi _N(A),\hat\phi _N(B))\le 2(\varepsilon /4+\varepsilon /4) =\varepsilon \). The second statement follows from Lemma 35.6.1.8.

On each pair window, extend the orthonormal family \(\omega _j\) to a unitary mapping the encoded sector basis to that family. On each physical block, extend the polar isometry on \(S\) using its sequential factorization. The first layer has constant depth and the second has depth at most a constant times \(L\). Their product gives (26); linearity gives the superposition identity.

Circuit inversion and composition give (27). The seed equation gives \(Ua'=b'\). Unitary invariance yields \(\varepsilon (Ua,b')=\varepsilon (a,a')\); the error triangle bound gives (28).

Apply Theorem 35.7.2 and add the three depth bounds.

A product operator with identity factors outside \(S\) has an entry containing \(\delta _{\sigma _k,\tau _k}=0\) at the differing site \(k\). Extend this equality by linearity. A computational-basis vector is the product of its one-site basis vectors, so disjoint expectations factor.

Expand the expectation difference as \(\langle \psi -\phi |A\psi \rangle + \langle \phi |A(\psi -\phi )\rangle \) and apply Cauchy–Schwarz. Expand the difference of the two products of expectations into two terms. The three expectation differences each contribute at most \(2\| \psi -\phi \| \).

The unobserved site kills both cross terms. Within each product branch, \(\langle XY\rangle _s=x_sy_s\). Subtract the product of the two weighted expectations and use \(|a|^2+|b|^2=1\). Each diagonal expectation has modulus at most one. For a circuit, pull both observables back through the unitary, retaining their support inside the backward light cones.

A boundary matrix \(X\) for \(A\) contributes amplitudes \((X_{00}+X_{11},X_{22})\); one for \(B\) contributes \((X_{00},X_{11})\). Both pairs range over \(\mathbb C^2\). For the identity boundary they become \((2,1)\) and \((1,1)\), whose norms on a positive chain are \(\sqrt5\) and \(\sqrt2\). The local orthogonal projectors are determined by these boundary-generated spaces. The same local terms define the chain constraints and Hamiltonians.

Choose sites \(i=0\), \(j=2T+2\) and \(k=T+1\). The radius-\(T\) backward cones of \(i,j\) are disjoint, and neither contains \(k\), including around the periodic boundary. The Pauli observables \(Z_i,Z_j\) have covariance at most \(16/25\) in modulus in \(U\psi _N\), while their covariance in \(\phi _N\) is one. Choose \(|c|=1\) with \(\eta ^2=\| U\psi _N-c\phi _N\| ^2 =2(1-|\langle U\psi _N|\phi _N\rangle |)\). Correlation stability gives \(1\le 16/25+6\eta \), hence \(\eta \ge 3/50\) and the stated bound.

Write \(P=(V_{A,N}^\dagger V_{A,N})^{1/2}\). The polar identities \(F_{A,N}=V_{A,N}G_{A,N}^{-1/2}\) and \(V_{A,N}=F_{A,N}P\) give both range inclusions. Injectivity makes the initial support projector \(I_b\).

Reading any one register recovers its word, so distinct labels give distinct physical basis configurations. The corresponding columns of the identity matrix are orthonormal.

Theorem 35.7.1 chooses one circuit whose action agrees on every column of \(J\). Therefore \(UJ=W\) as matrices, and \(W^\dagger W=J^\dagger U^\dagger UJ=I_b\).

Expand matrix multiplication and the Euclidean inner product in the common physical configuration basis.

The square-root Lipschitz bound at \(I\) gives the first inequality. Since \(V=F(V^\dagger V)^{1/2}\) and \(\| F\| \le 1\), it follows that \(\| F-V\| \le \eta \). Moreover, \((V-W)^\dagger (V-W)=(V^\dagger V-I)-(W^\dagger V-I)-(W^\dagger V-I)^\dagger \), so \(\| V-W\| ^2\le \eta +2\zeta \). Add these two distances.

Inversion and composition give the depth. Since \(U(U_AJ)=U_BJ\), the difference is \(U(F_A-U_AJ)+(U_BJ-F_B)\). Use the triangle inequality and \(\| U\| =1\).

Both \(U_AJ\) and \(U_BJ\) are isometries. Apply Theorem 35.8.7 to each family, use the monotonicity of the square root in (42), and then apply Theorem 35.8.8.

Expand the matrix entries. The pair identities follow by taking Kronecker products of the single-coordinate identities.

Induction on a nonempty word eliminates the cross terms using \(E_i^\dagger E_j=0\). Taking the trace gives the periodic identity; reading the bond-pair entries gives the physical-matrix identity.

Multiplying the blocked direct sum by \(K_j\) extracts the blocked matrix of \(A_j\). Summing its diagonal virtual entries takes its trace, giving the \(j\)th periodic column. The remaining identities follow directly from the coordinate indicators.

If \(V_{A,N}\beta =C_NT\beta =0\), the vector \(T\beta \) is supported on \(S\), so supported injectivity gives \(T\beta =0\). Choose one diagonal coordinate in each positive-dimensional sector. That coordinate of \(T\beta \) is \(\beta _j\), proving \(\beta =0\).

The polar identities give

Their positive factors are the same function of this Gram matrix, so taking polar partial isometries yields (45).

The identity \(S_NR_N=V_{A,N}\) makes \(R_N\) injective whenever \(V_{A,N}\) is. Its polar factor is therefore isometric. For support, let \(\Pi _S\) be the coordinate support projector. The identities \(\Pi _SP_N=P_N\) and \(R_N=P_NT\) imply \(\Pi _SR_N=R_N\). As \(Q_N\) is \(R_N\) times a function of its Gram matrix, \(\Pi _SQ_N=Q_N\) as well.

The cyclic pair on one block contributes \(\omega (r,l)=v(l,r)\). The product over one block and the sum over its virtual input reduce to the matrix-vector product \(S_Nv\).

Define \(\omega _j(l,r)=(Q_N)_{(r,l),j}\). These pair vectors are orthonormal, and their cyclic products are supported on the actual single block. The simultaneous pair unitary followed by the genuine sequential blocked-polar circuit produces \(S_NQ_Ne_j=F_{A,N}e_j\) for every \(j\), with depth at most \(CN\). All normalization is performed in the fixed space \(\mathbb {C}^{D^2}\); no arbitrary full-chain synthesis is used.

The diagonal entries are the squared sector norms minus one; the off-diagonal entries are complex periodic overlaps. Their existing transfer estimates give entry bounds \(K_{ij}x^N\). Sum these bounds weighted by the norms of the logical matrix units, over the fixed indices \(i,j\). No sum over the physical dimension occurs.

The exact cross-overlap identity identifies each entry with the trace of an ordered mixed-transfer product. Its difference from \(\delta _{ij}\) is bounded by \(C_{ij}y e^{C_{ij}y}\). Choose \(K_Z{\gt}0\) dominating all \(C_{ij}\) and their sum weighted by the norms of the logical matrix units. Sum the entry bounds with these weights. The sums again have only the fixed logical dimension.

Both defects are bounded respectively by \(K_Gz\) and \(K_Zze^{K_Z}\). Apply the polar-frame estimate and use \(0\le z\le \sqrt z\) for \(0\le z\le 1\).

Use the simultaneous genuine-unitary block construction for the orthonormal embedded fixed-point pairs. Supported injectivity gives independent endpoint columns. The common transfer bound with \(\gamma =1/2\) supplies the two defects in the preceding estimate. The exact assertion is the fixed-width whole-encoder construction. Increase the constants to cover both depth bounds.

Take \(q=\lceil a\log (N/\varepsilon )+b\rceil \), with \(a=2/r\) and \(b=\max (\log K,0)/(r/2)+L+1\). Set \(u=Ne^{-(r/2)q}\). The logarithmic threshold gives \(Ku\le \varepsilon \), while \(Ne^{-rq}\le u^2\) because \(N\ge 1\). Thus the square-root error is at most \(Ku\le \varepsilon \), and the small-defect hypothesis holds. If \(q{\gt}N\), use the exact conversion. In either branch, the depth is at most \(Cq\), and \(q\le c_0\log (N/\varepsilon )\) for a fixed \(c_0\), using \(\log (N/\varepsilon )\ge \log 2\).

Choose one common strict spectral bound, one register width large enough for both virtual bonds and the logical labels, and one eventual support-injectivity cutoff. Both approximate circuits use the same partition and register encoder \(J\). Their relative unitary \(U_BU_A^\dagger \) has the sum of their depths and endpoint errors. The exact one-block circuits share the same seed as well. Apply the preceding all-accuracy theorem. The operator error tensored with an identity has the same bound, by summing squared norms of reference slices. No preparation of the common seed is charged.

The polar encoder has exactly the periodic-sector span. The canonical bounded-range parent theorem identifies this span with the kernel at the stated interaction range, including \(N=L\).

The squared norm \(\lVert \phi _M(VP_\infty )\rVert ^2\) is the sum of the squared moduli of its coefficients, which equals \(1\) by the norm identity recorded with Definition 35.3.8.

\(|\hat\phi _N(B)\rangle =(\zeta ^N/\lvert \zeta ^N\rvert )|\hat\phi _N(A)\rangle \), and if \(|\phi _N(A)\rangle =0\) both normalized states vanish.

Theorem 35.5.7 at \(\gamma =\xi /(2a){\lt}1\) gives \(K\) with error at most \(KMe^{-q/a}\); take \(b_0=a\max (\log K,0)\) and apply Lemma 35.5.4. The error against \(\phi _N(A)\) equals that against \(\phi _N(B)\) by Lemma 35.8.2.9.

Since \(a{\gt}\xi /2\), the number \(\gamma =\xi /(2a)\) satisfies \(0{\lt}\gamma {\lt}1\), and Theorem 35.5.7 at this \(\gamma \) gives \(K\) with \(\varepsilon (\widetilde\phi _N,\phi _N(B))\le K\, (N/q)\, e^{-q/a}\), since \(2\gamma /\xi =1/a\). Take \(b=a\max (\log K,0)+L_0+3D+1\), with \(L_0\) a length at which the blocked tensor is injective. Then \(q\ge b\) makes the blocked tensor \(\mathcal B_q\) injective by Theorem 6.1.1.24 and Lemma 6.5.3, and \(q\ge 3D\). Moreover \(q/a\ge \log (N/\varepsilon )+\log K\ge \log (N/(q\varepsilon ))+\log K\), so \(K(N/q)e^{-q/a}\le \varepsilon \). The approximating state is a unit vector by Lemma 35.8.2.8, is prepared in depth at most \(Cq\) by Theorem 35.6.1, and has the same error against \(\phi _N(A)\) as against \(\phi _N(B)\) by Lemma 35.8.2.9.

The transfer map is a channel whose only eigenvalue of modulus \(1\) is \(1\), and it has finitely many eigenvalues, so the others are at most \(1-\delta \) in modulus for some \(\delta {\gt}0\); take \(t=\max (1-\delta ,1/2)\).

Since \(N/\varepsilon \geq 2\), \(\log (N/\varepsilon )\geq \log 2\), so \(b\leq (b/\log 2)\log (N/\varepsilon )\), and \(Cq\leq 2C(a\log (N/\varepsilon )+b)\leq C(2a+2b/\log 2)\log (N/\varepsilon )\).

A gauge transformation and a rescaling bring \(A\) into the gauge (3) by Theorem 12.8.8, and the positive definite fixed point \(\rho \) it provides is replaced by \(\sigma =\rho /\operatorname{Tr}\rho \), still a fixed point of trace one, by Theorem 6.6.2; they multiply \(|\phi _N(A)\rangle \) by a nonzero scalar. The new tensor is normal by Theorems 12.8.1 and 2.5.7. By Theorem 7.6.3 its transfer map is primitive, so \(1\) is its only eigenvalue of modulus \(1\). The gauge is left canonical, so the transfer map is a channel by Theorem 2.2.8, and all its eigenvalues have modulus at most \(1\) by Theorem 5.3.1, and finitely many of them give a bound \(0{\lt}t{\lt}1\) on the moduli of the other eigenvalues. With \(\xi =-1/\log t{\gt}0\), Theorem 35.8.2.11 at the slope \(a=\xi {\gt}\xi /2\) gives \(b\ge 1\) and the first statement. For the second statement, Lemma 35.8.2.13 gives \(Cq\le C(2a+2b/\log 2)\log (N/\varepsilon )\).

A word of length \(L_1+L_2\) is the concatenation \(ww'\) of words of lengths \(L_1\) and \(L_2\), with \(A^{ww'}=A^wA^{w'}\) and \(\langle ww'|X\otimes Y|vv'\rangle =\langle w|X|v\rangle \langle w'|Y|v'\rangle \); summing gives \(\mathcal{E}_{X\otimes Y}=\mathcal{E}_X\mathcal{E}_Y\), and \(\mathcal{E}_{\mathbb {1}}=\mathcal{E}_A^{\, L}\) follows by induction on \(L\). The adjoint formula is the conjugate of the defining sum. Writing \(\langle w | \phi _N(A) \rangle =\operatorname{Tr}A^w\) and expanding \(\sum _{w,w'}\overline{\operatorname{Tr}A^{w'}}\langle w'|O|w\rangle \operatorname{Tr}A^w\) in matrix units gives the trace of \(\mathcal{E}_O\).

Choose coefficients \(c^{ab}_w\) with \(\sum _wc^{ab}_wA^w=e_{ab}\). The operator with matrix elements \(\langle w'|X|w\rangle =c^{ab}_w\overline{c^{ce}_{w'}}\) gives \(\mathcal{E}_X(Z)=e_{ab}Ze_{ec}=Z_{be}\, e_{ac}\). These maps span all linear maps on \(M_{D}(\mathbb {C})\), and \(X\mapsto \mathcal{E}_X\) is linear. If \(\Phi \) preserves adjoints and \(\mathcal{E}_X=\Phi \), then by Lemma 35.8.2.15 also \(\mathcal{E}_{X^\dagger }=\Phi \), so the Hermitian part \((X+X^\dagger )/2\) has the same map.

An operator on a window that does not wrap around the ring is the tensor product of the identity on the sites before it, the operator on the window, and the identity on the sites after it. Hence \(X_1Y_{L+m+1}\) is \(X\otimes \mathbb {1}\otimes Y\otimes \mathbb {1}\) over the blocks of lengths \(L,m,L,n\), and Lemma 35.8.2.15 gives the trace formula.

Trace preservation gives \(\lambda \operatorname{Tr}R=\operatorname{Tr}R\), so \(\operatorname{Tr}R=0\), \(P(R)=0\), and \((\mathcal{E}_A-P)^tR=\lambda ^tR\). By Lemma 35.8.2.16 take \(\mathcal{E}_Y(Z)=\operatorname{Tr}(Z)\, R\) and \(\mathcal{E}_X(Z)=(Z_{ij}/R_{ij})\, \rho \) for an entry \(R_{ij}\neq 0\).

The Vandermonde map \(v\mapsto (\sum _jv_j\mu _j^{\, u})_{0\le u{\lt}K}\) is injective, so it is bounded below on \(\mathbb {C}^K\). The vector \((a_j\mu _j^t)_j\) has the maximum norm of \(a\).

Since \(\langle Z\rangle _\phi \) is real, \((\langle Z\rangle _\phi -\langle Z\rangle _\psi )\langle \phi | \psi \rangle =\langle \phi |(Z-\langle Z\rangle _\psi )|\psi \rangle -\langle \phi |(Z-\langle Z\rangle _\phi )|\psi \rangle \); apply the Cauchy–Schwarz inequality to both terms.

This is the second and third paragraph of the proof of Theorem 35.8.2.23: a traceless Hermitian vector \(v\) and a functional \(\ell \) with \(\ell (Z^\dagger )=\overline{\ell (Z)}\) are built from an eigenvector \(R\) and its adjoint, the maps \(Z\mapsto \operatorname{Tr}(Z)\, v\) and \(Z\mapsto \ell (Z)\, \rho \) are realized by Hermitian observables through Lemma 35.8.2.16, and scaling to norm one multiplies \(a\) by a positive constant.

A nonzero eigenvalue of \(\mathcal{E}_A-P\) is an eigenvalue of \(\mathcal{E}_A\) with a traceless eigenvector, hence different from \(1\) by strong irreducibility (Theorem 6.1.1.15), so the spectral radius of \(\mathcal{E}_A-P\) is at most \(\lvert \lambda _2\rvert \) and Gelfand’s formula gives (1). Part (2) substitutes \(\mathcal{E}_A^{\, k}=P+(\mathcal{E}_A-P)^k\) for \(k\ge 1\) into the trace formula of Lemma 35.8.2.17 and into the one-point functions and the normalization. Part (3) is elementary algebra with \(\lvert 1/z\rvert \le 2\), and part (4) expands the trace in matrix units and uses \(\xi =-1/\log \lvert \lambda _2\rvert \).

With the inserted transfer maps \(\mathcal{E}_X\) defined before Lemma 35.8.2.16, and for operators \(X,Y\) on consecutive blocks,

Expanding the traces in (1) gives, for \(L{\lt}s\) and \(s+L-1\le N\),

the trace of a linear map on \(M_{D}(\mathbb {C})\). Let \(P(Z)=\operatorname{Tr}(Z)\, \rho \). In the gauge (3), \(\mathcal{E}_A^{\, n}=P+(\mathcal{E}_A-P)^n\) for \(n\ge 1\). An eigenvalue \(\nu \neq 0\) of \(\mathcal{E}_A-P\) is an eigenvalue of \(\mathcal{E}_A\) with a traceless eigenvector, so \(\nu \neq 1\) because \(\mathcal{E}_A\) is strongly irreducible (Theorem 6.1.1.15), and hence \(\lvert \nu \rvert \le \lvert \lambda _2\rvert \). Fix \(r\) with \(\lvert \lambda _2\rvert {\lt}r{\lt}1\) and \(r^2{\lt}\lvert \lambda _2\rvert \); Gelfand’s formula gives \(\lVert (\mathcal{E}_A-P)^n\rVert \le C_rr^n\). Replacing the last power \(\mathcal{E}_A^{\, N-s-L+1}\) by \(P\), and likewise in the normalization \(\operatorname {tr}\mathcal{E}_A^{\, N}\) and in the one-point functions, changes \(G_N(X,Y;s)\) by at most \(C_{X,Y}\, r^{N-s-L+1}\), which is at most \(C_{X,Y}\, r^{2(s-L-1)}\) when \(N\ge 3s\). The remaining term is

Let \(R\neq 0\) satisfy \(\mathcal{E}_A(R)=\lambda _2R\). Since \(\xi {\gt}0\), \(\lambda _2\neq 0\), and trace preservation gives \(\operatorname{Tr}R=0\), so \((\mathcal{E}_A-P)^tR=\lambda _2^{\, t}R\); since \(\mathcal{E}_A(Z^\dagger )=\mathcal{E}_A(Z)^\dagger \), also \((\mathcal{E}_A-P)^tR^\dagger =\bar\lambda _2^{\, t}R^\dagger \). If \(\lambda _2\) is real, one of \(R+R^\dagger \) and \(i(R-R^\dagger )\) is a nonzero Hermitian eigenvector \(H\); put \(v=H\) and \(\ell (Z)=\operatorname{Tr}(HZ)/\operatorname{Tr}(H^2)\), so that \(\ell ((\mathcal{E}_A-P)^tv)=\lambda _2^{\, t}\). If \(\lambda _2\) is not real, \(R\) and \(R^\dagger \) are eigenvectors for distinct eigenvalues; take a linear functional \(f\) with \(f(R)=1\) and \(f(R^\dagger )=0\), put \(v=R+R^\dagger \) and \(\ell (Z)=f(Z)+\overline{f(Z^\dagger )}\), so that \(\ell ((\mathcal{E}_A-P)^tv)=\lambda _2^{\, t}+\bar\lambda _2^{\, t}\). In both cases \(v\) is Hermitian, \(\ell (Z^\dagger )=\overline{\ell (Z)}\), and \(\ell ((\mathcal{E}_A-P)^tv)=\lvert \lambda _2\rvert ^t\sum _{j=1}^Ka_j\mu _j^t\) with \(K\le 2\), distinct \(\mu _j\) of modulus one and \(a\neq 0\).

The maps \(Z\mapsto \operatorname{Tr}(Z)\, v\) and \(Z\mapsto \ell (Z)\, \rho \) preserve Hermitian matrices. By Lemma 35.8.2.16 they are \(\mathcal{E}_Y\) and \(\mathcal{E}_X\) for some \(X,Y\), and since \(\mathcal{E}_{X^\dagger }(Z)=\mathcal{E}_X(Z^\dagger )^\dagger \) the Hermitian parts of \(X\) and \(Y\) have the same maps. Then \(G_\infty (X,Y;s)=\lvert \lambda _2\rvert ^t\sum _ja_j\mu _j^t\) with \(t=s-L-1\), and \(X,Y\neq 0\) because this sum does not vanish identically. Take \(\mathcal O,\mathcal O'\) to be \(X,Y\) scaled to norm one; this multiplies \(a\) by a positive constant.

The Vandermonde matrix \(W=(\mu _j^{\, u})_{0\le u{\lt}K,\, 1\le j\le K}\) is invertible because the \(\mu _j\) are distinct. For every \(t\) the vector with entries \(\sum _ja_j\mu _j^{\, t+u}\), \(0\le u{\lt}K\), is \(W\) applied to \((a_j\mu _j^t)_j\), whose maximum norm equals that of \(a\) because \(\lvert \mu _j\rvert =1\). Hence some \(u\in \{ 0,\dots ,K-1\} \) satisfies

so \(\lvert G_\infty (\mathcal O,\mathcal O';s+u)\rvert \ge c_0\lvert \lambda _2\rvert ^{t'}\) with \(t'=t+u\). With \(s'=s+u\) and \(N\ge 3s'\) the finite-size correction is at most \(C\, r^{2t'}=C\, (r^2/\lvert \lambda _2\rvert )^{t'}\lvert \lambda _2\rvert ^{t'}\), which is at most \(\tfrac {c_0}{2}\lvert \lambda _2\rvert ^{t'}\) after enlarging \(s_0\), since \(r^2{\lt}\lvert \lambda _2\rvert \). Finally \(\lvert \lambda _2\rvert ^{t'}=e^{-(s'-L-1)/\xi }\ge e^{-(s'-1)/\xi }\), which gives (47) with \(c=c_0/2\).

We may assume that \(A\) is in the gauge (3). Since \(A\) is normal, \(\mathcal{E}_A\) is strongly irreducible (Theorem 6.1.1.15), so its spectral radius \(\lambda _1\) is a positive eigenvalue whose right eigenvector \(\rho \) and whose eigenvector \(\sigma \) for the dual map are positive definite. Replacing \(A^i\) by \(\lambda _1^{-1/2}\sigma ^{1/2}A^i\sigma ^{-1/2}\) gives (3), with fixed point \(\sigma ^{1/2}\rho \sigma ^{1/2}\) scaled to unit trace. The scaling multiplies \(|\phi _N\rangle \) by a positive constant and the conjugation leaves every trace in (1) unchanged, so \(\phi _N/\lVert \phi _N\rVert \) is unchanged; the eigenvalues of \(\mathcal{E}_A\) are divided by \(\lambda _1\), which is the normalization \(\lambda _1=1\) in which \(\xi \) is computed.

Take \(L,\mathcal O,\mathcal O',c,K,s_0\) from Theorem 35.8.2.23, and suppose \(\varepsilon (\phi _N,\psi _N)\le 1/2\), that is \(\lvert \langle \phi _N | \psi _N \rangle \rvert \ge 1/2\). Write \(T=T_N\) and \(|\psi _N\rangle =U|\pi \rangle \) with \(U\) a local circuit of depth \(T\) and \(|\pi \rangle \) a product state. If \(Z\) acts on an interval of sites, then \(U^\dagger ZU\) acts on the interval enlarged by \(T\) sites on each side, so expectations in \(|\psi _N\rangle \) of products of operators whose enlarged intervals are pairwise disjoint factorize.

Let \(s_N=\max (s_0,2T+L+1)\) and let \(s'\in \{ s_N,\dots ,s_N+K-1\} \) be given by the lemma; the window changes \(s'\) by at most \(K-1\), so still \(s'=o(\log N)\) and \(N\ge 3s'\) for large \(N\). Put \(\Delta =s'+L+2T\le 2s'\), \(n=\lfloor N/\Delta \rfloor \ge N/(4s')\), and \(b_k=1+(k-1)\Delta \) for \(k=1,\dots ,n\). With \(e=\langle \phi _N|\mathcal O_1|\phi _N\rangle \) and \(e'=\langle \phi _N|\mathcal O'_1|\phi _N\rangle \) let \(\mathcal O^{(k)}\) be \(\mathcal O-e\) on the sites \(b_k,\dots ,b_k+L-1\) and \(\mathcal O'^{(k)}\) be \(\mathcal O'-e'\) on the sites \(b_k+s'-1,\dots ,b_k+s'+L-2\). These have norm at most \(2\), zero expectation in \(|\phi _N\rangle \) by translation invariance, and \(\langle \phi _N|\mathcal O^{(k)}\mathcal O'^{(k)}|\phi _N\rangle =G_N(\mathcal O,\mathcal O';s')\). Since \(s'\ge 2T+L+1\) and \(\Delta \ge s'+L+2T\), the \(2n\) enlarged intervals are pairwise disjoint, also around the ring.

Two estimates are used. For Hermitian \(Z\) and unit vectors \(\phi ,\psi \), with \(\sigma _\chi (Z)=\lVert (Z-\langle Z\rangle _\chi )\chi \rVert \),

because \((\langle Z\rangle _\phi -\langle Z\rangle _\psi )\langle \phi | \psi \rangle =\langle \phi |(Z-\langle Z\rangle _\psi )|\psi \rangle -\langle \phi |(Z-\langle Z\rangle _\phi )|\psi \rangle \). Secondly, the decomposition \(\mathcal{E}_A^{\, n}=P+(\mathcal{E}_A-P)^n\) from the proof of Theorem 35.8.2.23, together with the fact that in the gauge (3) the maps \(\mathcal{E}_Z\) have norm bounded by a multiple of \(\lVert Z\rVert \) independent of the size of the support, shows that the connected correlation in \(|\phi _N\rangle \) of two operators of norm at most \(4\) on intervals separated by \(g\) and \(g'\) sites around the ring is at most \(C(r^g+r^{g'})\), with \(C\) and \(r{\lt}1\) depending only on \(A\). Consecutive intervals \([b_k,b_k+s'+L-2]\) are separated by at least \(2T\) sites, and intervals \(j\) apart by at least \((j-1)\Delta \), so every average of \(n\) of these operators, or of their pair products, has variance at most \(C_1/n\) in \(|\phi _N\rangle \).

For signs \(\epsilon _k\in \{ \pm 1\} \) apply (50) to \(Z=\frac1n\sum _k\epsilon _k\mathcal O^{(k)}\). Its expectation in \(|\phi _N\rangle \) is \(0\); in \(|\psi _N\rangle \) it is \(\frac1n\sum _k\epsilon _kx_k\) with \(x_k=\langle \psi _N|\mathcal O^{(k)}|\psi _N\rangle \in \mathbb {R}\), and by factorization its variance there is at most \(4/n\). Hence \(\lvert \frac1n\sum _k\epsilon _kx_k\rvert \le C_2n^{-1/2}\), and the choice \(\epsilon _k=\operatorname {sign}x_k\) gives \(\frac1n\sum _k\lvert x_k\rvert \le C_2n^{-1/2}\). Next apply (50) to \(Z=\frac1n\sum _k\mathcal O^{(k)}\mathcal O'^{(k)}\), which is Hermitian because the two factors act on disjoint sites. Its expectation in \(|\phi _N\rangle \) is \(G_N(\mathcal O,\mathcal O';s')\). In \(|\psi _N\rangle \) the light cone gives \(\langle \psi _N|\mathcal O^{(k)}\mathcal O'^{(k)}|\psi _N\rangle =x_ky_k\) with \(\lvert y_k\rvert \le 2\), so the expectation is at most \(2C_2n^{-1/2}\) in modulus and the variance is at most \(16/n\). With (47) and \(n\ge N/(4s')\),

Since \(s'\le s_N+K-1=o(\log N)\), the left side is \(N^{-o(1)}\) and the right side is \(N^{-1/2+o(1)}\), so (51) fails for all \(N\) larger than some \(N_0\) determined by \(A\) and the sequence \((T_N)\). For those \(N\), \(\varepsilon (\phi _N,\psi _N){\gt}1/2\).

The argument is quantitative. With the overlap bound \(1/2\) replaced by \(\eta \in (0,1]\), the constants \(C_2\) and \(C_3\) grow at most like \(\eta ^{-1}\). Since \(s'\le 2T+C\), and the bound below is trivial when \(T\ge \xi \ln N/4\), (51) shows that \(\lvert \langle \phi _N | \psi _N \rangle \rvert \ge \eta \) requires

with \(C_A\) depending only on \(A\); for \(\eta =1/2\) it follows from the form (53) of Theorem 35.8.2.28. The same constant \(\xi /4\) in front of \(\ln N\) is implicit in the final inequality of the proof in [ 219 ] , which compares \(N/\ln N\) with a multiple of \(e^{4T/\xi }\).

Part (1) is the trace formula of Lemma 35.8.2.17 at an arbitrary offset: the exponent of \(\mathcal{E}_A\) closing the ring changes by the offset and the trace absorbs it. For (2), the entry of \(\mathcal{E}_X\) applied to a matrix unit is \(\langle p|X|q\rangle \) for vectors \(p,q\) of entries of products of matrices of \(A\), and \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\) makes their norms at most one. Part (3) substitutes \(\mathcal{E}_A^{\, k}=P+(\mathcal{E}_A-P)^k\) into the trace formulas, as in Lemma 35.8.2.22. For (4) and (5), \(\lVert (Z-\langle Z\rangle )\chi \rVert ^2\) is the double sum of \(c_kc_l\) times the connected correlations of \(X^{(k)}\) and \(X^{(l)}\); in \(|\phi _N\rangle \) these are bounded by (3) with \(g\ge l-k\) and \(g'\ge n-l+k\), and in \(|\psi \rangle \) they vanish for \(k\ne l\) by Lemma 33.1.7, since the windows are at least \(2T\) sites apart on both sides.

Part (1) is the argument of the proof of Theorem 35.8.2.24 from its second paragraph on, with \(s'\le s_0+2T+L+2\), \(\Delta \le (s_0+2L+4)(T+1)\) and \(n\ge N/(2\Delta )\). Squaring (51) gives \(N\le C_1\Delta \, e^{2(s'-1)/\xi }\), and \(e^{2(s'-1)/\xi }\le e^{2(s_0+L+1)/\xi }e^{4T/\xi }\), which is (1). For (2), put \(b=4/\xi \); \(T_N\le \log N/(2(b+1))\) for large \(N\) gives \(C(T_N+1)e^{b(T_N+1)}\le Ce^{b+1}\sqrt N{\lt}N\) and \(B(T_N+1)\le N\), so \(\lvert \langle \phi _N | \psi _N \rangle \rvert {\lt}1/2\) by (1). For (3), if \(N\ge B(T+1)\) take logarithms in (1); otherwise \(\ln N\le \ln (B+1)+\ln (T+1)\). In both cases \(\ln N\le \lvert \ln C\rvert +\ln (B+1)+\ln (T+1)+4T/\xi \); multiply by \(\xi /4\).

The eigenvalues of \(\mathcal{E}_{\zeta A}=\lvert \zeta \rvert ^2\mathcal{E}_A\) are those of \(\mathcal{E}_A\) times \(\lvert \zeta \rvert ^2\), and a gauge \(B^i=XA^iX^{-1}\) maps an eigenvector \(Y\) of \(\mathcal{E}_A\) to the eigenvector \(XYX^\dagger \) of \(\mathcal{E}_B\) with the same eigenvalue; it leaves every trace in (1) unchanged. Since \(A\) is normal, \(\mathcal{E}_A\) is irreducible, so it has a positive eigenvector for its spectral radius \(r{\gt}0\) and a positive eigenvector \(\sigma \) of its dual; with \(\zeta =r^{-1/2}\) and \(X=\sigma ^{1/2}\), the tensor \(B\) satisfies \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\) (Theorem 8.2.3). By Theorem 6.1.1.16, \(\mathcal{E}_B\) has a fixed point \(\rho {\gt}0\), which we scale to unit trace, and \(1\) is its only eigenvalue of modulus one. All eigenvalues of the trace-preserving map \(\mathcal{E}_B\) have modulus at most one, and \(\lvert \zeta \rvert ^2\lambda _1\) is one of largest modulus, so \(\lvert \zeta \rvert ^2\lambda _1=1\). When \(A\) is irreducible and \(1\) is the unique eigenvalue of \(\mathcal{E}_A\) of largest modulus, \(r=1\), so \(\zeta =1\); primitivity of \(\mathcal{E}_A\) passes to \(\mathcal{E}_B\), and some blocking of \(B\) is injective by Theorems 8.2.4 and 2.5.7.

By Lemma 35.8.2.27, \(A\) may be replaced by a tensor in the gauge (3); this changes \(|\phi _N\rangle \) by a phase and divides the eigenvalues of \(\mathcal{E}_A\) by \(\lambda _1\), so \(\xi \) and \(\lvert \langle \phi _N | \psi \rangle \rvert \) are unchanged. Part (1) is then part (3) of Theorem 35.8.2.26. For (2) put \(\theta =4a/\xi {\lt}1\) and \(l=(1-\theta )/a\). Multiplying (53) by \(\theta \) gives \(a\ln N\le \theta T+a\ln (T+1)+\theta C\), and \(\ln x=\ln (lx)-\ln l\le lx-1-\ln l\) gives \(a\ln (T+1)\le (1-\theta )(T+1)-a(1+\ln l)\). Adding, \(a\ln N\le T+C_a\) with \(C_a=1-\theta -a(1+\ln l)+\theta C\).

For (1), Theorem 35.6.1.7 gives depth at most \(c_0\ln (N/\varepsilon )\) for \(N\ge N_0\), and for \(N\ge 2\), \(\ln (N/\varepsilon )=\ln N+\ln (1/\varepsilon ) \le \bigl(1+\ln (1/\varepsilon )/\ln 2\bigr)\ln N\). Part (2) is part (2) of Theorem 35.8.2.28, since \(\varepsilon \le 1/2\).