Put \(d=Vx\, Vy-V(xy)\). Expanding \(\tau _A(d^*d)\) gives four terms. The reflected four-point identity evaluates the first, the three-point identity evaluates the two cross terms, and the two-point identity evaluates the last. After preserving involutions and associating products, their values are all \(\tau _B(y^*x^*xy)\), with coefficients \(1,-1,-1,1\). Hence \(\tau _A(d^*d)=0\), and quadratic faithfulness gives \(d=0\).

The first assertion is the tensor-extension closure theorem. For the second, let \(\tau _H:R\otimes H\simeq H\otimes R\) and \(\tau _K:K\otimes R\simeq R\otimes K\) be the canonical factor swaps. Then

where \(\mathcal R_\tau \) denotes reindexing both matrix coordinates along \(\tau \). Both reindexing maps are trace-preserving and completely positive, so closure under composition proves the claim.

Composing each Kraus family of \(\Phi _i\) with the corresponding \(P_i\) gives one combined rectangular Kraus family representing the sum.

Trace preservation of each \(\Phi _i\) reduces the left side to \(\sum _i\operatorname{tr}(P_iXP_i)\); cyclicity of the trace and idempotence of each projection give \(\operatorname{tr}(P_iXP_i)=\operatorname{tr}(P_iX)\).

For every pair of physical words \(\sigma ,\tau \),

Under these identifications, the length-one and length-two word evaluations are \(M^{ij}\) and \(M^{i_0j_0}M^{i_1j_1}\), respectively.

The length-three word evaluation is \(M^{i_0j_0}M^{i_1j_1}M^{i_2j_2}\).

The blocking relation \(A^{i_1}A^{i_2}=\sum _jV_{(i_1,i_2),j}A^j\) implies

For \(d{\gt}0\), the refinement identity follows directly:

For the coarse-graining identity, put \(P=VV^\dagger \). Since \(V^\dagger V=\mathbb {1}\) and \((\mathbb {1}-P)V=0\), substitution gives

If \(d=0\), the unique zero channels satisfy both identities. Conversely, trace preservation of \(\mathcal T\) gives idempotence of the physical-trace transfer of \(M(A)\), as in [ 13 , Appendix C, lines 1333–1340 ] . After exchanging the two tensor factors, this matrix is the transfer matrix of \(\mathcal{E}_A\). Faithfulness of the transfer-matrix representation therefore gives \(\mathcal{E}_A^2=\mathcal{E}_A\). The physical-isometry formulation follows from Theorem 17.1.6.

Multiply the row indexed by \(\alpha \) in \(B^n\) by the two columns of \(B\).

Expand \(B^{N+2}=B^{N+1}B\) and \(B^{N+1}=B^NB\). Substitution of \(B=\left(\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right)\) into the diagonal sums gives \(a_{N+2}+a_N=3a_{N+1}\).

Substitute \(x^1=(1,1,1,2)\) into the recurrence once and close the two diagonal boundary labels.

Use two-step induction on \(N\). The cases \(N=0\) and \(N=1\) follow by direct calculation. Since \(\tau _\pm ^2=\tau _\pm +1\), each sequence \(\tau _\pm ^{2N}\) satisfies \(\tau _\pm ^{2(N+2)}+\tau _\pm ^{2N}=3\tau _\pm ^{2(N+1)}\). The induction step now follows from the same recurrence for \(a_N\).

Such a formula would give \(r=a_1=3\) and \(7=a_2=3s\), which is impossible in \(\mathbb {N}\).

A diagonal entry is nonzero exactly when its physical labels form an allowed path from \(\alpha \) to \(\beta \). Splitting a path at its first physical label shows that the numbers of supported words obey the transition rule determined by \(B\), with the same length-zero initial values as \(x^n_{\alpha \beta }\). Induction therefore identifies the number of nonzero diagonal entries with \(x^n_{\alpha \beta }\). The rank of a diagonal matrix is the number of its nonzero diagonal entries.

The diagonal MPO rank counts physical words with nonzero periodic amplitude. For \(N{\gt}0\), the supports obtained by closing the virtual endpoint at \(0\) and at \(1\) are disjoint, since the first physical bit determines the initial endpoint. Their cardinalities are \(x^N_{00}\) and \(x^N_{11}\) by the open-boundary formula. The golden-ratio identity for \(a_N\) gives the final equality.

The operator rank equals \(a_N\) at every positive length, while \(a_1=3\) and \(a_2=7\). A geometric formula would therefore give \(r=3\) and \(7=3s\), which is impossible in \(\mathbb {N}\).

Suppose otherwise, and let \(P\) be the positive witness for the strong fixed-point relation. Then, for every \(N{\gt}0\), the periodic ranks satisfy

(8) contradicts the non-geometricity of the Fibonacci periodic ranks.

All four identities hold by definition of the product-index equivalences.

If \((K_j)_j\) are Kraus operators for \(\mathcal T\) with \(\sum _j K_j^\dagger K_j=I_d\), then the localized map has Kraus operators \(K_j\otimes I_{d^N}\), and

The two regroupings are unitary conjugations and preserve complete positivity and the trace. The argument for \(\mathcal S\) is identical.

For words \(u,v\) on the \(N\) spectator sites, write \(M^{u,v}\) for their virtual word evaluation and write \([Y]^{(r)}_{u,v}\) for the block of an operator \(Y\) on the first \(r\) sites. Regrouping the first site gives

Therefore

Equality of all blocks proves the refinement identity. Interchanging \(M_1\), \(M_2\) and using \(\mathcal S[M_2(Y)]=M_1(Y)\) proves the coarsening identity. Setting \(X=1\) and using \(M_N(1)=\rho ^{(N)}(M)\) (Lemma 20.1.6) gives the periodic-ring identities

One direction follows by taking \(N=1\). For the other, apply the localized refinement channel repeatedly. By Theorem 20.2.21, it sends \(\rho ^{(N)}(M)\) to \(\rho ^{(N+1)}(M)\), and complete positivity preserves each matrix inequality.

A ket–bra word evaluation vanishes at the first unequal pair of physical labels. Equal words recover the original MPS word evaluation. The rank formula follows by counting the nonzero entries of a diagonal matrix.

In the double sum defining \(\mathcal{E}_{A^{\mathrm{MPO}}}\), all off-diagonal terms vanish because \(M^{ij}=0\) for \(i\ne j\), while the diagonal terms are exactly \(A^iX(A^i)^\dagger \). Thus only the sum over \(i\) remains, which is the defining formula for \(\mathcal{E}_A\).

Both conditions assert idempotence of the corresponding transfer map. Theorem 20.3.3 identifies those transfer maps, so the two predicates are equivalent.

Combine Theorem 20.3.4 with the characterization of zero correlation length by idempotence of the MPS transfer map (Theorem 17.5.20).

Expand the contractions over the common physical and inward vertical indices, and decode the two packed bond coordinates. This gives (9).

Identify each MPO bond coordinate with a pair of horizontal PEPS virtual coordinates. Substituting the displayed purifying matrices into the LPDO contraction and summing over the ancillary pair \((s,b)\) gives exactly (9).

Apply the general implication from local purification to matrix product density operators.

The domain and codomain of the physical isometry, the virtual-space identifications, the transition from the finite-cylinder operators \(\sigma _L,\sigma _R\) to the local tensor, and the normalization of the refinement channel are unspecified in [ 13 ] ; complete positivity, trace preservation, and the intertwining identities \(\mathcal S[M_2(X)]=M_1(X)\) and \(\mathcal T[M_1(X)]=M_2(X)\) await complete algebraic formulation [ 136 ] .

Multiplying (10) on the right by \(X\) and taking the virtual trace gives, entrywise,

which is (11). Conversely, subtract the right-hand side of (10). By (11), the trace of the product with every \(X\) vanishes, so nondegeneracy of the trace pairing makes the difference zero.

At row \(((i,p),u)\) and column \(((j,q),v)\), both sides equal \(Y_{(i,u),(j,v)}P_{p,q}\).

Apply Theorem 20.2.21 to the local map \(Y\mapsto U(Y\otimes P)U^\dagger \) and the identity (11). This gives the localized closure equality. Expanding the regroupings \(R_1\) and \(R_2\), localized unitary conjugation is conjugation by \(U\otimes \mathbb {1}\), while Lemma 20.5.4 gives the middle factor \(\Sigma _N\! \left(R_1[M_{N+1}(X)]\otimes P\right)\). The identity closure from Lemma 20.1.6 gives the periodic formula.

Left and right multiplication by invertible matrices preserve rank.

Unitary conjugation and relabelling of matrix coordinates preserve rank, while rank is multiplicative under Kronecker products. Applying these facts to (15) gives the recurrence. Induction from chain length one gives the geometric formula. Finally, Theorem 20.5.2 exposes the same positive witness \(P\) for any tensor satisfying the existential strong fixed-point relation.

This is the defining equation.

Idempotence gives \(\mathcal T_M^r=\mathcal T_M\) for every \(r{\gt}0\). Apply this identity to the wrapping power in each side of (20); the middle power is unchanged.

Idempotence gives \(\mathcal T_M^r=\mathcal T_M\) for every \(r{\gt}0\). Applying this identity to both powers in each instance of (19) proves (21).

On a zero-dimensional bond space every endomorphism vanishes, contrary to the nonzero physical-trace transfer in Definition 20.6.7.

This is Definition 20.6.7 with \(\lambda =1\).

Choose \(\lambda {\gt}0\) from the up-to-scalar relation, so that \(\mathcal T_M^2=\lambda \, \mathcal T_M\). Then

By Lemma 16.2.2, the transfer map of \(M\) agrees with the transfer map of its doubled-index MPS view. Thus \(\mathcal{E}_M\circ \mathcal{E}_M=\mathcal{E}_M\) holds exactly when the transfer map of that doubled-index MPS tensor is idempotent, which is the RFP condition.

For every virtual matrix \(X\), the one-site and two-site closures satisfy

Write \(\Phi \) for the one-to-two trace-preserving map \(\mathcal T\) of Definition 20.2.1; this is distinct from the physical-trace transfer \(\mathcal T_M\). Then \(\Phi [M_1(X)]=M_2(X)\), and hence

Nondegeneracy of the trace pairing now gives \(\mathcal T_M^2=\mathcal T_M\).

Since the two virtual sectors are diagonal, set \(z_0=1\) and \(z_1=-1\). A physical configuration \(\sigma \) has weight

This proves the closed-operator formula. Each diagonal entry is either \(2^{-N}+4^{-N}\) or \(2^{-N}-4^{-N}\) and is nonnegative. Contracting the physical indices gives \(\mathcal T_K=\operatorname {diag}(1,0)\), whence \(\operatorname{tr}\rho _N(K)=\operatorname{tr}(\mathcal T_K^N)=1\) for \(N\geq 1\) and \(\mathcal T_K^2=\mathcal T_K\).

Direct contraction with a virtual matrix \(X\) gives the two closure formulas. Set \(X_* = \operatorname {diag}(2,8)\). Then

If the coarse-graining channel \(\mathcal S\) of Definition 20.2.1 existed, complete positivity would give \(\mathcal S[K_2(X_*)]\geq 0\), whereas its intertwining identity would give \(\mathcal S[K_2(X_*)]=K_1(X_*)\). This is a contradiction.

Every letter of \(A_K\) is diagonal and the block letters are scalars, so the weighted direct sum is the entrywise check

whose right-hand sides are the two virtual diagonal entries of \(K^{ii}\); the letters with \(i\ne j\) vanish on both sides. Each block satisfies \(\sum _{i,j}A_k^{(i,j)}\, \overline{A_k^{(i,j)}}=1\), hence has identity transfer map and is a normal tensor by Lemma E.1.3. The weighted direct sum of positive-dimensional normal blocks is in literal CPSV canonical form by Definition 7.6.7.

Since \(\operatorname{tr}\rho _N(K)=1\), the normalized state is \(\rho _N(K)\) itself. For \(1\leq L{\lt}N\), the partial trace over the complementary \(N-L\) sites annihilates the summand \(4^{-N}Z^{\otimes N}\) of \(\rho _N(K)=2^{-N}I+4^{-N}Z^{\otimes N}\), since \(\operatorname{tr}Z=0\), and contracts the remaining summand to \(2^{-L}I\). Hence \(\rho _L^{(N)}(K)=2^{-L}I\) and \(S_L^{(N)}(K)=L\log 2\). Substitution in \(I_L^{(N)}=S_L^{(N)}(K)+S_{N-L}^{(N)}(K)-S_N^{(N)}(K)\) gives \(I_L^{(N)}=N\log 2-S_N^{(N)}(K)\), which is independent of \(L\); in particular \(I_L=I_{L+1}\) whenever \(1\leq L{\lt}\lfloor N/2\rfloor \), so \(K\) saturates the area law by Definition 20.8.4.

Theorem 20.6.14 supplies the remaining three properties: \(K\) generates an MPDO, its physical-trace transfer is idempotent, and it is not a renormalization fixed point. Together with canonical form and area-law saturation, this proves the claim.

Apply Theorem 20.6.13 and take the positive scalar in Definition 20.6.7 to be \(1\).

This is Lemma  [ 216 , “The right partial trace is trace-preserving completely positive” ] with \(A\) the spin space and \(B\) the ancillary space.

Forget the renormalization-fixed-point condition on the purifying tensor.

Apply Theorem 16.3.3 to the local purification supplied by Theorem 20.7.9.

Take the diagonal purification at \(d=d_K=2\) and \(D=D'=1\) with amplitudes \(A=[\tfrac {1}{\sqrt2},0,0,\tfrac {1}{\sqrt2}]\). The purifying tensor is a pure-state renormalization fixed point, since \(\sum |A|^2=1\) makes its transfer map the identity. The ancilla contraction \(M^{ij}=\sum _k A^{(i,k)} \overline{A^{(j,k)}}\) gives the scalar entries \(M^{00}=M^{11}=\tfrac 12\) (off-diagonal \(0\)), so the induced transfer map is \(\tfrac 12\cdot \operatorname{id}\) and \(\mathcal{E}_M\circ \mathcal{E}_M=\tfrac 14\cdot \operatorname{id}\neq \mathcal{E}_M\). The trace contraction has dropped the leading eigenvalue from \(1\) to \(\tfrac 12\). By contrast, the physical-trace transfer is \(1\) and hence is literally idempotent. Thus this example separates doubled-index idempotence from the paper’s physical-trace condition.

This is the direct computation of the two nonzero diagonal entries of the tensor.

Lemma 20.7.12 gives \(\mathcal T_{M_{\mathrm w}}=1\). The identity is nonzero and idempotent, so the relation holds with \(\lambda =1\).

Let \(K'\) be the transfer matrix of \(\widehat A\), let \(s\) interchange the two Kronecker factors, and let \(K\) denote the same Kronecker sum in the physical ordering. The local contraction gives

Since both reindexings are bijective and commute with matrix multiplication,

Injectivity of the transfer-matrix representation gives

Apply the forward implication of Theorem 20.7.14 to the purifying tensor and bond identification in the PRFP presentation.

The forward direction combines Theorems 20.7.9 and 20.7.15. Conversely, choose the local purifying tensor and bond identification and apply Theorem 20.7.14 to that fixed purification.

Combine Theorem 20.7.16 with the nonzero clause in Definition 20.7.8.

Theorem 20.7.15 gives literal idempotence, and the nonzero hypothesis gives the scalar-one case of Definition 20.6.7.

Apply Theorem 20.7.18 to the two clauses of Definition 20.7.8.

Applying the mixed-product property \((A\otimes B)(C\otimes D)=(AC)\otimes (BD)\) repeatedly, the length-\(N\) product of the contracted tensor entries distributes over the ancillary index sum:

Tracing this sum and using \(\operatorname{tr}(X\otimes \overline Y)=\operatorname{tr}(X) \overline{\operatorname{tr}(Y)}\) gives the coefficient of the ancillary trace at every \(\sigma ,\tau \). The equation holds at every length, so no positive-length restriction is needed.

The local ancillary-contraction identity implies equality of the two generated density operators at every length by Theorem 20.7.20.

The spectral decomposition of the Hermitian reduced state identifies the characteristic roots with its eigenvalues. The von Neumann entropy is the sum of \(-r\log r\) over these eigenvalues, counted with multiplicity.

Compose the successive equalities from the smaller index to the larger.

The defining condition supplies \(I_m=I_{m+1}\) throughout the interval \(1\le m{\lt}\lfloor N/2\rfloor \). Apply Lemma 20.8.5.

Write \(\rho ^{(k)}\) for the first-\(k\)-site reduced state, with \(\cong \) denoting equality up to a permutation of basis indices. Reindexing a matrix does not change its von Neumann entropy, giving \(S(ABC)=S_{a+b+c}\) immediately. Tracing out \(C\) leaves the first \(a+b\) sites, \(\operatorname{tr}_C(\rho _{ABC})\cong \rho ^{(a+b)}\), so \(S(AB)=S_{a+b}\). Tracing out \(A\) and \(C\) leaves the middle \(b\) sites, while tracing out \(A\) leaves the final \(b+c\) sites; translation invariance gives

Hence \(S(B)=S_b\) and \(S(BC)=S_{b+c}\).

Apply strong subadditivity to the reduced state of the first \(N-L\) spins, split into three contiguous segments of lengths \(1\), \(L\), and \(N-2L-1\). Tracing the appropriate segments yields the four block entropies \(S_{N-L}\), \(S_L\), \(S_{L+1}\), and \(S_{N-L-1}\), where the cyclic invariance of \(\rho ^{(N)}(M)\) identifies the entropy of a contiguous block with that of any block of the same length. Strong subadditivity then gives \(S_{N-L}+S_L\le S_{L+1}+S_{N-L-1}\), which rearranges to \(I_L\le I_{L+1}\) after subtracting \(S_N\).

A contraction around the virtual loop vanishes unless the bra and ket configurations agree and are both constant. The all-zero configuration receives weight one. The all-one configuration receives the sum of the two virtual-sector weights, namely \(1+(-1)^N\). This gives the displayed diagonal form. Its coefficients are non-negative: the second coefficient is zero for odd \(N\) and two for even \(N\). Hence every positive-length operator is positive semidefinite, and summing its two possible diagonal entries gives the trace \(2+(-1)^N\).

After normalization, the all-zero weight is consequently one at odd lengths and \(1/3\) at positive even lengths. Tracing out all but the first site does not change this entry, since among the configurations beginning with zero only the all-zero configuration has nonzero weight. The two subsequences of one-site marginals therefore have distinct all-zero entries, so the sequence cannot converge.

At odd \(N\), the normalized state and both of its nonempty marginals are pure, so all three terms in \(I_L^{(N)}=S_L+S_{N-L}-S_N\) vanish. At even \(N\), the full state and both marginals have the same two nonzero eigenvalues \(1/3\) and \(2/3\). Each of their entropies is therefore \(h_2(1/3)\), and hence \(I_L^{(N)}=h_2(1/3)+h_2(1/3)-h_2(1/3)=h_2(1/3)\). The binary entropy is strictly positive at \(1/3\). Thus the odd and even subsequences have distinct constant values.

The MPDO property follows from positivity of every positive-length periodic operator. The formulas of Theorem 20.8.12 yield two subsequences with distinct values \(0\) and \(h_2(1/3){\gt}0\) at every fixed cut \(L\ge 1\), so no limit exists.

The first marginal is the defining partial trace over the last \(N-L\) sites. Periodic translation invariance identifies the second marginal with the reduced state on the first \(N-L\) sites. Entropy is invariant under the finite-index identifications, and retaining the full chain leaves the normalized state unchanged.

Fixing the word on the second interval leaves a physical closure on the first interval, whose virtual boundary is the fixed word evaluation. Apply the appropriate global closure identity to this slice. Fixing the resulting word on the first interval gives the closure on the second interval. Cyclicity of the virtual trace identifies the two cuts of the periodic contraction. The reverse identity is the same argument with \(\mathcal S\) and \(\mathcal T\) interchanged.

Apply mutual-information data processing to the forward pair of local channels and to the reverse pair. The two transfer identities turn the resulting inequalities into opposite inequalities between the displayed mutual informations.

Normalized BNT-refined horizontal form gives a nonzero sector compression for one positive length. Positivity makes its trace nonzero. The fixed-point equations make the physical-trace transfer idempotent, so the trace of its positive powers is independent of the positive exponent.

Write the cut as \((a+1):(b+2)\). The two channel identities and data processing give \(I(\sigma ^{(N)}_{a+2:b+1})\leq I(\sigma ^{(N)}_{a+1:b+2})\) and \(I(\sigma ^{(N)}_{a+1:b+2})\leq I(\sigma ^{(N)}_{a+2:b+1})\). Hence the two quantities are equal. The consecutive-cut identification gives \(I_L(M)=I_{L+1}(M)\).

Lemma 20.8.18 gives nonzero trace at every positive length. Apply Theorem 20.8.19.

Trace preservation in the local renormalization equations gives \(\mathcal T_M^2=\mathcal T_M\) for the physical-trace transfer. If \(\mathcal T_M=0\), then \(\operatorname{tr}[\rho ^{(1)}(M)]=\operatorname{tr}(\mathcal T_M)=0\), contradicting nonvanishing at length one. Thus the literal idempotence is nontrivial, and Theorem 20.8.19 gives saturation of the area law.

Lemma 20.8.18 gives nonzero trace at every positive length. Apply Theorem 20.8.21.

The reduced state of a block is a density matrix, and the von Neumann entropy of a density matrix is non-negative.

Apply strong subadditivity with a trivial middle subsystem (equivalently, subadditivity) to the bipartition of the chain into the first \(L\) and last \(N-L\) spins: \(S_N+S_0\le S_L+S_{N-L}\). Since \(S_0\ge 0\) (Lemma 20.8.23), rearranging gives \(I_L=S_L+S_{N-L}-S_N\ge S_0\ge 0\).

Both sides equal \(S_L+S_{N-L}-S_N\) by the symmetric definition \(I_L=S_L+S_{N-L}-S_N\), using \(N-(N-L)=L\).

If its rank were zero, every eigenvalue would vanish, contradicting that their sum is the trace, which equals one.

Its rank is positive by Lemma 20.8.26, hence it is exactly one. The rank bound gives \(S\le \log 1=0\), while entropy is non-negative.

The reduced state of zero spins is one-dimensional, so its rank is at most one. Lemma 20.8.27 gives \(S_0=0\).

By definition, \(I_0=S_0+S_N-S_N=S_0\), which is zero by Lemma 20.8.28.

The reindexing equivalence is a channel, as is the partial trace over the ancillary factor. Their composition is therefore trace-preserving and completely positive.

After regrouping the spin and ancillary factors, the definition of the partial trace gives the first coefficient identity. Applying this identity to both tensor factors gives the displayed iterated sum; the two finite sums may be interchanged.

The coefficient formula for one block gives

Trace preservation gives equality of the two normalizing traces. Splitting \(\kappa \) into its left and right restrictions changes the last sum into the two independent sums in Theorem 20.8.32, which proves the normalized identity across \(L\mid K\).

Choose one local-purification representation of \(M\). For an arbitrary decomposition \(N=L+K\), apply Theorem 20.8.33 to the same purifying tensor.

Write \(\sigma _A^{(N)}\) for the normalized pure state of \(A\) in the \(L\mid (N-L)\) bipartition. Then

The equality is the channel-image hypothesis, the first inequality is data processing, and the last inequality is Proposition 20.10.23.

By Theorem 20.8.33,

The first inequality is data processing and the second is the pure-state boundary estimate. Nonvanishing of the purifying pure state implies \(D'{\gt}0\). For an LPDO, positivity follows from Theorem 16.3.3, while the trace identity in Theorem 20.8.33 transfers the nonzero-trace hypothesis to the purifying pure state.

For each word, write \(A_x=M^{x_0x_0}\cdots M^{x_{L-1}x_{L-1}}\) and \(B_y=M^{y_0y_0}\cdots M^{y_{R-1}y_{R-1}}\). The identity

factors \(P\) through the \(D^2\)-dimensional space indexed by pairs \((a,b)\).

For every finite selection of columns, the selected columns are linearly independent over \(K\) if and only if their scalar extensions are linearly independent over \(L\). Selecting a basis from each column space proves the two rank inequalities.

Normalization by the nonzero scalar \(Z^{-1}\) does not change real rank. Faithful scalar extension does not change rank by Theorem 20.8.38; hence Theorem 20.8.37 gives \(\operatorname{rank}_{\mathbb {R}}P\leq D^2\). Applying [ 216 , “Classical mutual information is bounded by ordinary rank” ] gives \(I(X:Y)_P\leq \log \operatorname{rank}_{\mathbb {R}}P\), and monotonicity of the logarithm gives the result.

Pairing each configuration \(\sigma \) on \(L+R\) sites with its unique block split \((x,y)\) on \(L\) and \(R\) sites gives

Positive semidefiniteness makes every diagonal entry real and non-negative, so \(W_{x,y}\geq 0\) and its scalar extension recovers the complex diagonal cut matrix. Since \(Z\) is the sum of the entries of \(W\), \(\sum _{x,y}\widehat P_{x,y} =Z^{-1}\sum _{x,y}W_{x,y}=1\). Thus \(\widehat P\) is a joint probability distribution. Diagonality says that these coefficients constitute the full classical finite-chain state, and Theorem 20.8.39 gives the bound.

Global diagonality and positivity give the fixed-length hypotheses of Theorem 20.8.42. Since the chain operator is diagonal,

so the trace positivity hypothesis supplies the required normalization.

Split each closed virtual word into its left and right products. Expanding the trace over the two exposed virtual indices gives the displayed sum of \(D^2\) product matrices. The minimality property of the operator-Schmidt rank then bounds it by the number of displayed terms.

Reindex the normalized chain operator across the consecutive cut. Its trace is one, and its positivity is preserved by this reindexing. Hence

Nonzero trace makes the normalized operator nonzero, so its operator-Schmidt rank is positive. If \(D=0\), then every finite-chain trace vanishes. Hence \(D{\gt}0\), which justifies the last logarithmic identity.

Global positivity supplies positivity at the chosen positive chain length. Apply Theorem 20.10.1. The same nonzero-trace argument gives \(D{\gt}0\), hence \(0\leq \log D\) and \(2\log D\leq 4\log D\).

The word evaluation of the doubled tensor is the Kronecker product of the word evaluation of \(A\) and its conjugate, \(M^{\sigma ,\tau }=A^\sigma \otimes \overline{A^\tau }\), so its trace factors as \(\operatorname{tr}(M^{\sigma ,\tau }) =\operatorname{tr}(A^\sigma ) \overline{\operatorname{tr}(A^\tau )} =\overline{V^{(N)}(A)_\tau }\, V^{(N)}(A)_\sigma \), the corresponding matrix element of \(|V^{(N)}(A)\rangle \! \langle V^{(N)}(A)|\).

Lemma 20.10.6 gives \(\rho ^{(N)}(M)=|V^{(N)}(A)\rangle \! \langle V^{(N)}(A)|\), hence

Therefore the reduced block states agree:

Thus \(S_L^{(N)}(M) =S(\rho _L^{(N)}(M)) =S(\rho _L^{(N)}(A)) =S_L^{(N)}(A)\).

The reduced state of the first \(L\) spins is \(\rho _L\propto WW^\dagger \), where \(W\) is the matrix of amplitudes \(\langle u\, w|V^{(N)}(A)\rangle \) with \(u\) the first \(L\) spins and \(w\) the remaining \(N-L\). By cyclic invariance of the amplitudes, the reduced state of the complement is \(\propto \overline{W^\dagger W}\). Cyclic invariance of the entropy, \(S(WW^\dagger )=S(W^\dagger W)\), together with invariance of the entropy under entrywise conjugation of a Hermitian matrix, \(S(\overline{W^\dagger W})=S(W^\dagger W)\), gives \(S_L^{(N)}(A)=S_{N-L}^{(N)}(A)\).

Apply the block-entropy form of strong subadditivity, the inequality underlying Proposition 20.8.9, to the purification MPDO of \(A\), giving \(S_{N-L}+S_L\le S_{L+1}+S_{N-L-1}\). By the Schmidt symmetry \(S_{N-L}=S_L\) and \(S_{N-L-1}=S_{L+1}\), this reduces to \(S_L\le S_{L+1}\).

The left Gram entry is \(\sum _\sigma \overline{(A^\sigma )_{a_1a_2}}(A^\sigma )_{b_1b_2}\) by definition of \(L_\ell \). Expanding

and reading off the \((b_1,a_1)\) entry gives the same sum, since

The right-factor formula is the same expansion with the two boundary bonds interchanged: the \((p,q)\) entry of \(R_mR_m^\dagger \) is \(\sum _\tau (A^\tau )_{p_2p_1}\overline{(A^\tau )_{q_2q_1}}\), which is the \((p_2,q_2)\) entry of \(\mathcal{E}_A^{\, m}(|p_1\rangle \! \langle q_1|)\).

By definition, a renormalization fixed point satisfies \(\mathcal{E}_A^{\, 2}=\mathcal{E}_A\), so \(\mathcal{E}_A^{\, k}=\mathcal{E}_A\) for every \(k\ge 1\). Applying Lemma 20.10.10 to the left and right Gram entries gives the two displayed identities.

The reduced block state has the exact factorization

Cyclic invariance of the charpoly-root entropy sum gives

which is the displayed formula.

Write \(c=(\operatorname{tr}\rho ^{(N)})^{-1}\). Lemma 20.10.12 computes the entropy from the bond environment:

Since \(1\le L{\lt}\lfloor N/2\rfloor \), the four lengths \(L\), \(L+1\), \(N-L\), and \(N-(L+1)\) are positive. Lemma 20.10.11 gives

Therefore the two bond environments agree:

The scalar \(c\) depends only on \(N\), so the two charpoly-root entropy sums are equal. Hence \(S_L^{(N)}(A)=S_{L+1}^{(N)}(A)\).

The fixed point saturates the area law (Theorem 20.10.13), so by the telescoping of saturation (Lemma 20.10.21) the block entropies at any two lengths \(1\le L,L'\le \lfloor N/2\rfloor \) agree: \(S_L^{(N)}(A)=S_{L'}^{(N)}(A)\).

The reduced state is \(\rho _L\propto WW^\dagger \), where the amplitude matrix

factors through the two bond pairs cut by the block boundary. Writing the trace as a sum over the bond index pair \((a,b)\in \{ 1,\ldots ,D\} ^2\) exhibits \(W=L'R'\) with \(L'\) having \(D^2\) columns, so \(\operatorname{rank}\rho _L\leq \operatorname{rank}W\leq D^2\).

Positive semidefiniteness is preserved by the partial trace of the positive semidefinite normalized pure state, and the partial trace preserves the trace, which the normalization fixes to one.

The reduced state \(\rho _L^{(N)}(A)\) is a density matrix (Lemma 20.10.16), so its rank is positive by Lemma 20.8.26. Its rank is at most \(D^2\) by Lemma 20.10.15. The rank bound on the entropy gives \(S_L^{(N)}(A)\leq \log (D^2)=2\log D\).

The reduced state \(\rho _L^{(N)}(A)\) is a density matrix (Lemma 20.10.16), and the von Neumann entropy of a density matrix is non-negative.

After reindexing the full block, the normalized operator is the rank-one pure state

A pure state has von Neumann entropy zero.

By the Schmidt symmetry \(S_0^{(N)}(A)=S_N^{(N)}(A)\), and the entropy of the full system vanishes for a pure state (Lemma 20.10.19).

The defining condition supplies \(S_m^{(N)}(A)=S_{m+1}^{(N)}(A)\) throughout the interval \(1\le m{\lt}\lfloor N/2\rfloor \). Apply Lemma 20.8.5.

By definition, \(I_L=S_L+S_{N-L}-S_N\) for the generated state. The global entropy of the pure state vanishes (Lemma 20.10.19), and the Schmidt symmetry (Lemma 20.10.8) gives \(S_{N-L}=S_L\), so \(I_L=2S_L\).

For a pure state, \(I_L=2S_L^{(N)}(A)\), since the global entropy \(S_N\) vanishes and a block and its complement share the same Schmidt spectrum. The block-entropy bound \(S_L^{(N)}(A)\leq 2\log D\) (Proposition 20.10.17) then gives \(I_L\leq 4\log D\).

The \((\sigma ,\rho )\) entry of \(W^\dagger W\) factors into the product of the corresponding ket and bra entries of \(V^\dagger V\), which gives (25). Isometric mixing of a Kraus family then yields (26). Conversely, each original matrix is recovered from the mixed family by \(A^\sigma =\sum _\tau \overline{W_{\tau ,\sigma }}A_V^\tau \); this proves the injectivity equivalence. The transfer-map identity and the equivalence between irreducibility of a Kraus family and its transfer map preserve normality without requiring a separate spectral argument.

At length \(N\), the periodic operator of \(\mathcal K_V\) is \(V^{\otimes N}\rho ^{(N)}(\mathcal K)(V^{\otimes N})^\dagger \). This proves positivity for arbitrary \(V\). Under the isometry hypothesis, traces and the mutual informations of every contiguous block agree in the two physical spaces, which gives the MPDO and area-law equivalences in (27). Contracting the two physical indices and using \(V^\dagger V=\mathbb {1}\) gives the equality of the matrices \(\mathcal T\).

Finally, let \(Z_{\mathcal K}\) stack all columns of all physical slices. Directly from the definition, \(Z_{\mathcal K_V}=VZ_{\mathcal K}R\), where \(R\) is the blockwise copy of \(V^\dagger \) and \(RR^\dagger =\mathbb {1}\). Hence \(Z_{\mathcal K_V}Z_{\mathcal K_V}^\dagger =VZ_{\mathcal K}Z_{\mathcal K}^\dagger V^\dagger \), and taking column-support projections proves the last identity in (27). A common virtual similarity distributes termwise through the physical sums, proving the gauge assertion.