D Canonical Form Reduction: Supporting Results
The canonical form reductions of Chapter 7 rely on the algebraic and spectral behavior of projection, gauge transformation, blocking, and cyclic-sector reindexing developed here.
Throughout, \(\mathcal{E}_A(X)=\sum _iA^iX(A^i)^\dagger \), \(A^{[p]}\) denotes blocking by \(p\), and \(V^{(N)}(A)_\sigma =\operatorname{tr}(A^{\sigma _1}\cdots A^{\sigma _N})\). Weighted direct sums are written \(\bigoplus _k\mu _kA_k\).
D.1 Similarities, strict splitting, and zero blocks
Similarity invariance and strict dimension decrease justify the induction used in Theorem 7.1.1.2; the final result removes blocks that vanish at every positive length.
If \(U\) is a unitary matrix, then \(A\) and \(U^\dagger A U\) generate the same MPV family.
By cyclicity of the trace, \(\operatorname{tr}((U^\dagger A^{i_1} U) \cdots (U^\dagger A^{i_N} U)) = \operatorname{tr}(A^{i_1} \cdots A^{i_N})\).
Under the hypotheses of Theorem 7.1.2, assume in addition that \(P\neq 0\) and \(P\neq \mathbb {1}\). Then there exist \(n,m\) with \(n+m=D\), \(n{\lt}D\), and \(m{\lt}D\), together with block tensors \(A_1,A_2\), such that \(A\) and \(A_1\oplus A_2\) generate the same MPV family in the sense of Definition 2.4.3.
Apply Theorem 7.1.2. In the diagonal form of \(P\), the \(1\)-eigenspace is nonempty because \(P\neq 0\), and the \(0\)-eigenspace is nonempty because \(P\neq \mathbb {1}\). Hence both block sizes are positive. Since \(n+m=D\), this implies \(n{\lt}D\) and \(m{\lt}D\).
Every MPS tensor \(A\) admits a finite family of nonzero irreducible blocks \((B_k)_{k=1}^r\), each with at least one nonzero Kraus operator and positive bond dimension. For every \(N\ge 1\) and every \(\sigma \),
Moreover, \(\sum _{k=1}^rD_k\le D\).
Apply the irreducible block decomposition (Theorem 7.1.1.2) and retain the blocks having a nonzero Kraus operator. For every all-zero irreducible block \(C_j\) and every \(N\ge 1\), \(V^{(N)}(C_j)_\sigma =0\), so (1) holds. The dimension inequality follows by restricting the structural identity \(\sum _jD_j=D\) to the retained subset.
D.2 Projector closure and isometric corner decompositions
Reducing orthogonal projections yield support isometries that compress tensors to corner blocks, preserving matrix product vectors across direct sums (supporting Theorems 7.7.3 and 7.7.5).
Suppose that \(A\) has invariant-projector closure and that \(P\) is an orthogonal projection satisfying \((\mathbb {1}-P)A^iP=0\) for all \(i\). Then \(P\) reduces every tensor matrix: \(PA^i=A^iP\) for all \(i\).
The lower-corner relation gives \((\mathbb {1}-P)A^i=(\mathbb {1}-P)A^i(\mathbb {1}-P)\). Applying projector closure to \(\mathbb {1}-P\) gives \(A^i(\mathbb {1}-P)=(\mathbb {1}-P)A^i(\mathbb {1}-P)\), hence the upper corner \(PA^i(\mathbb {1}-P)\) also vanishes. Therefore \(PA^i=PA^iP=A^iP\).
If \(A\) has invariant-projector closure and admits a nontrivial invariant projection, then it admits a nontrivial orthogonal projection commuting with every \(A^i\).
Every orthogonal projection \(P\) on \(\mathbb {C}^D\) factors through an isometry: there are \(n\in \mathbb {N}\) and \(V\in \mathbb {C}^{D\times n}\) such that
Since \(P\) is Hermitian and idempotent, its eigenvalues are \(0\) and \(1\). Writing \(n=\operatorname{tr}(P)\), unitary diagonalization gives
The matrix \(V\in \mathbb {C}^{D\times n}\) formed by the first \(n\) columns of \(U\) satisfies (2).
The rectangular Gram-equality criterion that turns \(B^\dagger B=A^\dagger A\) into a unitary \(U\) with \(B=UA\) is a generic algebra fact, proved in the companion quantum-channel volume [ 216 , “Rectangular Gram equality gives a unitary” ] .
Let \(A\) have invariant-projector closure, and let \(V\) be an isometry, \(V^\dagger V=\mathbb {1}\), whose range projection \(VV^\dagger \) commutes with every \(A^i\). Then the corner tensor \(B^i=V^\dagger A^iV\) again has invariant-projector closure.
Let \(Q\) be an orthogonal projection with \(QB^i=QB^iQ\) for all \(i\). Then \(VQV^\dagger \) is an orthogonal projection. Commutation of \(VV^\dagger \) with each \(A^i\) gives \(V^\dagger A^i=B^iV^\dagger \), so left-invariance of \(Q\) yields
Projector closure for \(A\) yields \(A^i(VQV^\dagger )=(VQV^\dagger )A^i(VQV^\dagger )\), and compressing both sides between \(V^\dagger \) and \(V\) returns \(B^iQ=QB^iQ\).
Let \(V\) satisfy \(A^iV=VB^i\) for all \(i\). Then, for every \(X\),
If moreover \(V^\dagger V=\mathbb {1}\), then every eigenvalue of \(\mathcal{E}_B\) is an eigenvalue of \(\mathcal{E}_A\).
Expanding the left-hand side of (3) termwise, the intertwining \(A^iV=VB^i\) and its adjoint \(V^\dagger (A^i)^\dagger =(B^i)^\dagger V^\dagger \) move both factors past \(V\) and \(V^\dagger \). For the eigenvalue statement, if \(\mathcal{E}_B(X)=\mu X\) with \(X\neq 0\), then \(\mathcal{E}_A(VXV^\dagger )=\mu VXV^\dagger \), and \(VXV^\dagger \neq 0\) because compression with the isometry gives \(V^\dagger (VXV^\dagger )V=X\).
Let \(V_1,\ldots ,V_r\) be isometries, \(V_k^\dagger V_k=\mathbb {1}\), with \(\sum _kV_kV_k^\dagger =\mathbb {1}\), and suppose \(A^iV_k=V_kB_k^i\) for some tensors \(B_1,\ldots ,B_r\). Then \(\mathcal{V}(A)=\mathcal{V}(\bigoplus _{k=1}^rB_k)\).
The intertwining relation propagates from letters to words, so \(A^wV_k=V_kB_k^w\) for every word \(w\). Inserting the partition of unity \(\sum _kV_kV_k^\dagger =\mathbb {1}\) into the trace and cycling gives
which is the matrix product vector of the direct sum.
Let \(E_k:H_k\to \bigoplus _lH_l\) be the canonical inclusion and let \(B=\bigoplus _lB_l\). For distinct \(k\) and \(\ell \), the corresponding inclusions have orthogonal ranges. The identities are
In particular, \(B\) commutes with the range projection \(E_kE_k^\dagger \).
The identities in (4) follow directly from the defining block-diagonal entries. The relations \(BE_k=E_kB_k\) and \(E_k^\dagger B=B_kE_k^\dagger \) imply that \(B\) commutes with \(E_kE_k^\dagger \).
Suppose that, after a change of coordinates, \(A\) is the dependent direct sum of tensors \(B_k\). If \(A\) has invariant-projector closure and no nontrivial periodic vectors, then every \(B_k\) has both properties. The assertion also applies to summands of dimension zero.
Conjugate the canonical inclusion of the \(k\)-th summand back to the original bond coordinates of \(A\). This is an isometry which intertwines \(B_k\) with \(A\) on both sides, and its range projection commutes with every letter of \(A\). Projector closure therefore descends by exact compression. A periodic vector of \(B_k\) would embed through the same isometry as a periodic vector of \(A\).
Suppose that isometries \(V_k\) satisfy
Then \(A\) and \(\bigoplus _k\mu _kB_k\) have the same matrix product vectors at every positive length.
Expand the first letter by the reconstruction formula in (5). In each summand, move the remaining word through \(V_k^\dagger \) using the intertwining relation. Cyclicity of the trace places \(V_k^\dagger V_k=\mathbb {1}\) next to the weighted block word, leaving precisely its closed-chain trace.
D.3 Perron gauges and preservation under conjugation
Positive Perron eigenpairs determine the spectral normalization and unital gauge transformations of irreducible blocks.
Let \(B\) be an irreducible MPS tensor of positive bond dimension, not all of whose matrices vanish. Then there are \(\rho {\gt}0\) and \(r{\gt}0\) with \(\mathcal{E}_B(\rho )=r\rho \).
Irreducibility of the tensor makes the transfer map an irreducible completely positive map, and a nonzero matrix makes it nonzero, so it does not annihilate any nonzero PSD matrix. The Perron–Frobenius theorem for positive maps (Theorem [ 216 , “Positive-eigenvalue specialization” ] ) gives a nonzero PSD eigenvector with eigenvalue \(r{\gt}0\), and irreducibility upgrades it to a positive-definite one ( [ 244 , Theorem 6.3(2)–(3) ] ).
Let \(B\) be an irreducible MPS tensor of positive bond dimension with \(\mathcal{E}_B(\rho )=r\rho \) for some \(\rho {\gt}0\) and \(r{\gt}0\), and suppose every eigenvalue of \(\mathcal{E}_B\) of modulus \(r\) equals \(r\). Then \(r^{-1/2}B\) is a normal tensor.
The rescaling identity \(\mathcal{E}_{r^{-1/2}B}(X)=r^{-1}\mathcal{E}_B(X)\) gives \(\mathcal{E}_{r^{-1/2}B}(\rho )=r^{-1}r\rho =\rho \), so \(\rho \) is a positive-definite fixed point of \(\mathcal{E}_{r^{-1/2}B}\). The rescaled map remains irreducible and completely positive, so Theorem [ 216 , “Spectral radius is the Perron eigenvalue” ] identifies its spectral radius with the eigenvalue one. The eigenvalues of modulus one of \(\mathcal{E}_{r^{-1/2}B}\) are the eigenvalues of \(\mathcal{E}_B\) of modulus \(r\), divided by \(r\); by hypothesis the only such eigenvalue is one, so the peripheral eigenvalue set is \(\{ 1\} \). Irreducibility is unchanged by a nonzero rescaling.
The support projection \(\pi \) of a positive semidefinite matrix \(\rho \) factors as \(\pi =\rho W\) for a spectral pseudo-inverse \(W\), witnessing that the range of \(\pi \) is the range of \(\rho \). For a rectangular matrix \(Y\), the support projection \(\pi \) of \(YY^\dagger \) fixes \(Y\), \(\pi Y=Y\); and whenever \(AY=YG\) for some matrix \(G\)—that is, \(A\) maps the range of \(Y\) into itself—the one-sided invariance \((\mathbb {1}-\pi )A\pi =0\) holds.
In the spectral decomposition \(\rho =U\operatorname{diag}(\lambda )U^\dagger \), take \(W=U\operatorname{diag}(\lambda ^{+})U^\dagger \), where \(\lambda ^{+}\) inverts the positive eigenvalues and vanishes on the kernel. Then \(\rho W=U\operatorname{diag}(\mathbf{1}_{\lambda {\gt}0})U^\dagger =\pi \). From \(\pi (YY^\dagger )=YY^\dagger \), the matrix \(((\mathbb {1}-\pi )Y)((\mathbb {1}-\pi )Y)^\dagger \) vanishes, hence \(\pi Y=Y\). Finally,
Let \(B\) be an irreducible tensor, let \(X\) be invertible, and let \(c\ne 0\). Then the rescaled conjugated tensor with letters \(cX^{-1}B^vX\) is irreducible. In particular, the rescaled unital gauge of an irreducible tensor is irreducible; this is the normalization of [ 13 , Section 2.3 ] applied to an irreducible corner.
An invariant orthogonal projection \(P\notin \{ 0,\mathbb {1}\} \) of the conjugated tensor makes the columns of \(Y=XP\) span a subspace invariant under every letter of \(B\): \(B^vY=YG_v\), where \(G_v=X^{-1}B^vXP\). The support projection of \(YY^\dagger \) is then a nontrivial invariant orthogonal projection of \(B\) by Lemma D.3.3: it is nonzero because \(Y\ne 0\), and it is not the identity because any nonzero vector in the kernel of \(P\) transports through \((X^\dagger )^{-1}\) to the kernel of \(YY^\dagger \). This contradicts irreducibility of \(B\).
Let \(B\) be an irreducible tensor with a positive definite transfer eigenvector, \(\mathcal{E}_B(\rho )=r\rho \), where \(\rho {\gt}0\) and \(r{\gt}0\). Then the rescaled unital gauge \(K^v=r^{-1/2}\rho ^{-1/2}B^v\rho ^{1/2}\) is a unital Kraus family, is an irreducible tensor with an irreducible transfer map, and its adjoint map has a positive definite fixed point. These are the hypotheses of the cyclic decomposition of the peripheral spectrum, [ 244 , Theorem 6.6 ] .
Unitality is Theorem 7.8.2; irreducibility is Theorem D.3.4. The adjoint of a unital family is trace-preserving, so its transfer map is a channel and has a positive semidefinite fixed point by the Cesàro argument; irreducibility of the adjoint map upgrades the fixed point to a positive definite one.
In the setting of Theorem D.3.5, every transfer eigenvalue \(\mu \) of \(B\) becomes the transfer eigenvalue \(\mu /r\) of the rescaled unital gauge: the gauge conjugates the transfer map by the congruence with \(\rho ^{1/2}\) and divides it by \(r\). In particular, the peripheral eigenvalues \(e^{2\pi iq/p}\) described in [ 13 , Section 2.3 ] arise from the transfer eigenvalues of modulus \(r\).
If \(\mathcal{E}_B(Z)=\mu Z\), then \(Z'=\rho ^{-1/2}Z\rho ^{-1/2}\) is nonzero and satisfies \(\mathcal{E}_K(Z')=(\mu /r)Z'\) by direct computation.
D.4 Blocking identities and primitive stability
Transfer-map powers, weighted direct sums, and matrix product vectors commute with blocking, enabling the common-length reduction of Theorem 7.3.1.4.
Blocking by a length \(p\) commutes with weighted direct sums up to the \(p\)th power of each weight: for nonzero weights \((\mu _k)\) and blocks \((B_k)\),
where \(\sim \) is equality of MPV families. When two block families are reblocked to a common length \(p=m_ke_k\), the \(p\)-blocked alphabet is identified with the \(e_k\)-fold tensor power of the \(m_k\)-blocked alphabet, so each sector reblocked through its own period agrees with the directly \(p\)-blocked block.
Let \((B_k)_{k=1}^r\) be a finite family of positive-bond-dimension MPS tensors, each admitting a blocking period \(p_k\) making its transfer map primitive. Then \(p=\operatorname{lcm}(p_1,\ldots ,p_r)\) is a common period: \(\mathcal{E}_{B_k^{[p]}}\) is primitive for every \(k\).
Each \(p_k\) divides \(p\), so primitivity of \(\mathcal{E}_{B_k^{[p_k]}}\) carries over to \(\mathcal{E}_{B_k^{[p]}}\) via the divisibility identity for transfer-map powers.
Let \((B_k)_{k=1}^r\) be a finite family of positive-bond-dimension, left-canonical, tensor-irreducible MPS tensors. Then there is a single blocking length \(p{\gt}0\) such that \(\mathcal{E}_{B_k^{[p]}}\) is primitive for every \(k\).
Let \(A\) be an MPS tensor and let \(c\in \mathbb {C}\). If
then, for every integer \(L\geq 0\),
The transfer map of \(A^{[L]}\) is \(\mathcal{E}_A^L\) (Theorem D.4.8). Therefore
which is the required identity.
Let \(E\) be a linear endomorphism of a finite-dimensional complex inner product space. Then \(E\) is primitive if and only if its adjoint \(E^\dagger \) is primitive. Equivalently, the eigenvalues \(\lambda \) of \(E\) with \(|\lambda |=1\) form the singleton \(\{ 1\} \) if and only if the same holds for \(E^\dagger \), because adjunction sends every eigenvalue to its complex conjugate.
Let \(A\) have positive bond dimension and let \(p_0,p\ge 1\) with \(p_0\mid p\). If \(\mathcal{E}_{A^{[p_0]}}\) is primitive, then \(\mathcal{E}_{A^{[p]}}\) is primitive.
If a finite matrix family satisfies
then every blocked family satisfies the same identity.
Expand the blocked matrices as products indexed by words and sum one letter at a time.
Let \(E\) be a channel with a positive-definite fixed point. If all nonzero positive semidefinite fixed points of \(E\) are proportional, then \(E\) is irreducible.
A nontrivial invariant projection would produce a positive semidefinite fixed point not proportional to the faithful one.
For every tensor \(A\) and every \(L\), \(\mathcal{E}_{A^{[L]}}=\mathcal{E}_A^{\, L}\).
A letter of \(A^{[L]}\) is a word \(w\) of length \(L\) with matrix \(A^w\), so \(\mathcal{E}_{A^{[L]}}(X)=\sum _{\lvert w\rvert =L}A^wX(A^w)^\dagger \), which is the \(L\)-fold iterate of \(\mathcal{E}_A\) by induction on \(L\).
Let \(A\) be left-canonical, tensor-irreducible, and have primitive transfer map. Then \(A\) has primitive fixed-point data \((\rho ,P_\rho ,N)\) with \(\rho {\gt}0\) and \(N^n\to 0\). For every blocking length \(P\), the tensor \(A^{[P]}\) is left-canonical, its transfer map is \(\mathcal{E}_A^P\), and \(\rho \) is fixed by this blocked map. For every \(n\ge 1\),
Finally, a channel with a positive-definite fixed point whose positive semidefinite fixed points are all proportional is irreducible as a map; irreducibility of a transfer map implies tensor irreducibility.
Peripheral primitivity, tensor irreducibility, and left-canonical normalization first give primitive fixed-point data, and tensor irreducibility makes its positive semidefinite fixed point positive definite. Blocking preserves the left-canonical identity, replaces the transfer map by its positive power (Theorem D.4.8), and therefore preserves the fixed point. The defining complementary-power identity gives (7). The channel fixed-point criterion then gives map irreducibility from uniqueness of positive semidefinite fixed points, and the final implication converts map irreducibility to tensor irreducibility.
Let \(A\) be left-canonical, irreducible, primitive, of positive bond dimension. For every \(P{\gt}0\), the blocked tensor \(A^{[P]}\) is irreducible: there is no orthogonal projection \(Q\) with \(Q\neq 0\), \(Q\neq \mathbb {1}\) such that \((\mathbb {1}-Q)(A^{[P]})^\alpha Q=0\) for every \(P\)-blocked index \(\alpha \).
Lemma D.4.9 gives primitive fixed-point data with \(\rho {\gt}0\). In particular, if \(P_\rho (X)=\operatorname{tr}(X)\rho /\operatorname{tr}(\rho )\) and \(N:=\mathcal{E}_A-P_\rho \), then
The primitive fixed-point theorem supplies this complementary-power convergence.
The blocked transfer map is the corresponding positive power:
Hence \(\rho \) remains a positive-definite fixed point after blocking. Let \(\sigma \ge 0\) be any fixed point of \(\mathcal{E}_A^P\), set
Then \(\operatorname{tr}(\sigma ')=0\), so \(P_\rho (\sigma ')=0\), while \(\mathcal{E}_A^P(\sigma ')=\sigma '\). Since the fixed-point projection and the complement annihilate one another, iteration gives, for every \(k\ge 1\),
But \(N^{Pk}(\sigma ')\to 0\) as \(k\to \infty \), hence \(\sigma '=0\). Thus every positive semidefinite fixed point of the blocked map is proportional to \(\rho \).
Blocking preserves the left-canonical identity, so Theorem 2.2.8 makes the blocked transfer map a channel. Its positive-definite fixed point and the uniqueness just proved imply map irreducibility by Lemma D.4.9. The final implication in that lemma then shows that \(A^{[P]}\) is tensor-irreducible.
Let \(A\) have positive bond dimension and be trace-preserving, primitive, and tensor-irreducible. For every \(k{\gt}0\),
and \(A^{[k]}\) is tensor-irreducible. This \(k\) is a later common-refinement or injectivity length, distinct from the period-removal length that produced the sector block.
For nonzero weights \((\mu _k)_{k=1}^r\), blocks \((B_k)_{k=1}^r\), and any \(p\ge 1\), the blocking identity (6) holds, where \(\sim \) denotes generation of the same MPV family at every length.
If \(A\) generates the same MPV family as \(\bigoplus _k\mu _kB_k\) at every length, then for every \(p\ge 0\) the blocked tensor \(A^{[p]}\) generates the same MPV family as \(\bigoplus _k\mu _k^pB_k^{[p]}\) at every length.
For a blocked configuration \(\sigma \) of length \(N\) with unblocked configuration \(\sigma '\) of length \(pN\),
If two tensors have the same MPV coefficients at every positive length, then so do their blocked tensors after any positive blocking length.
If two tensors have eventually nonzero proportional MPV coefficients, then so do their blocked tensors after any positive blocking length.
If proportionality holds from length \(N_0\) onward, a blocked chain of length \(N\ge N_0\) expands to an original chain of length \(Np\ge N_0\). Use the original proportionality scalar at length \(Np\).
If \(A\) and \(\bigoplus _k\mu _kB_k\) have the same MPV coefficients at every positive length, then after any positive blocking length \(p\), \(A^{[p]}\) and \(\bigoplus _k\mu _k^{\, p}B_k^{[p]}\) have the same MPV coefficients at every positive length.
If \(\bigoplus _j\mu _j^AA_j\) and \(\bigoplus _k\mu _k^BB_k\) have the same MPV coefficients at every positive length, then after any positive common blocking length \(p\), \(\bigoplus _j(\mu _j^A)^pA_j^{[p]}\) and \(\bigoplus _k(\mu _k^B)^pB_k^{[p]}\) have the same MPV coefficients at every positive length.
If \(\mu \in \mathbb {C}\) is nonzero and \(p\ge 1\), then \(\mu ^p\neq 0\); blocking preserves the nonzero-weight hypothesis of a weighted block decomposition.
For any MPS tensor \(A\) and \(m,n\ge 0\),
D.5 Support compression and cyclic-sector families
Support isometries project a blocked tensor onto its cyclic corners, embedding multiple cyclic-sector families into a single common blocked alphabet.
Let \(A\) be a tensor normalized on the support of an orthogonal projection \(P\), so that \(PA^iP=A^i\) and \(\sum _i(A^i)^\dagger A^i=P\). Then there are a corner tensor \(C\), a linear isomorphism \(\varphi :M_{\dim C}(\mathbb {C})\xrightarrow {\sim }PM_{D}(\mathbb {C})P\), and a rectangular isometry \(V:\mathbb {C}^{\dim C}\to \mathbb {C}^D\) such that
Moreover, \(\varphi (C^i)=A^i\) for every physical letter, and for every word \(\sigma =(\sigma _1,\ldots ,\sigma _N)\),
Apply Theorem [ 216 , “Compression of a supported finite Kraus family” ] to the family \((A^i)_i\). It gives \(C\), \(\varphi \), and \(V\) satisfying (11) and \(\varphi (C^i)=A^i\). It remains to identify the coefficients of the associated words. For each word \(\sigma =(\sigma _1,\ldots ,\sigma _N)\), cyclicity of the trace and \(VV^\dagger =P\) give
which proves (12).
Let a left-canonical tensor \(A\) commute with a family of orthogonal projections \((P_k)\) summing to \(\mathbb {1}\). Then \(A\) decomposes into compressed sector tensors \((C_k)\), with a unit-weight direct-sum MPV decomposition, corner isomorphisms \(\varphi _k:M_{\dim C_k}(\mathbb {C})\xrightarrow {\sim }P_kM_{D}(\mathbb {C})P_k\), and rectangular isometries \(V_k\) satisfying
The sector letters obey \(\varphi _k(C_k^i)=P_kA^iP_k\).
Apply Theorem D.5.1 to the compression of \(A\) on each sector \(P_k\). Orthogonality and \(\sum _kP_k=\mathbb {1}\) give the direct-sum decomposition. For every word \(\sigma =(\sigma _1,\ldots ,\sigma _N)\), the partition of unity and (12) give
This is the unit-weight matrix-product-vector identity.
Let \(A\) be left-canonical and let \((P_k)\) be orthogonal projections summing to \(\mathbb {1}\) and fixed by the adjoint transfer map. Then the same sector decomposition and support isometries of Theorem D.5.2 exist. In particular, the adjoint fixed-point condition \(\mathcal{E}_A^\dagger (P_k)=P_k\) gives \(P_kA^i=A^iP_k\), which gives the corner-letter identity \(\varphi _k(C_k^i)=P_kA^iP_k\).
The fixed-point identity \(\mathcal{E}_A^\dagger (P_k)=P_k\), together with left-canonicality \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), places \(P_k\) in the multiplicative domain of \(\mathcal{E}_A^\dagger \), forcing \(P_kA^i=A^iP_k\) for every letter \(A^i\). The result then follows from Theorem D.5.2.
If \(A\) and \(B\) generate the same MPV family at every positive length, then after removing the all-zero blocks and applying the trace-preserving gauge there are weighted nonzero-block tensors \(\bigoplus _j\mu _j^AA_j\) and \(\bigoplus _k\mu _k^BB_k\) such that, for every \(N\ge 1\) and \(\sigma \),
The two nonzero parts agree:
Each nonzero block has primitive irreducible cyclic sectors: for each \(j\), there is \(m_j\ge 1\) with a unit-weight sector decomposition \(A_j^{[m_j]}\sim \bigoplus _{u=1}^{m_j}C^A_{j,u}\), and similarly for \(B_k\).
Given a finite family \((B_k)_{k=1}^r\), each with primitive irreducible cyclic sectors, there is a common blocking length \(p\) such that each \(B_k^{[p]}\) admits a unit-weight direct-sum decomposition over a single common blocked alphabet \(d^{[p]}\) into left-canonical blocks with primitive transfer maps, tensor irreducibility, and positive bond dimensions.
With hypotheses as in Lemma D.5.5, and given any common multiple \(p\) of the per-block period-removal lengths \((m_k)\), the common reblocking can be performed at the prescribed length \(p\).
Each sector tensor in the common reblocked cyclic-sector family of Lemma D.5.5 is trace-preserving, \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\), has primitive transfer map, \(\sigma _\partial (\mathcal{E}_B)=\{ 1\} \), is tensor-irreducible, and has positive bond dimension.
With \(p=m_ke_k\) for each \(k\), the iterated blocked tensor \((B_k^{[m_k]})^{[e_k]}\) agrees with \(B_k^{[p]}\) in MPV family, after the canonical identification of the \(p\)-blocked alphabet \(d^{[p]}\) with the \(e_k\)-fold tensor power of the \(m_k\)-blocked alphabet \(d^{[m_k]}\).
For every original block \(B_k\), direct blocking at the common length \(p=m_ke_k\) agrees, as an MPV family, with the same block transported to the common blocked alphabet. Write \(\widetilde{B}_k\) for this common-alphabet image. Then
The powered nonzero-part decomposition \(\bigoplus _k\mu _k^pB_k^{[p]}\) carries over under the common-alphabet identification of Lemma D.5.8: the common-alphabet sectors have the same nonzero-part decomposition as the direct \(p\)-blocked tensors.
D.6 Prepared normal-form consequences
For families of irreducible, trace-preserving, primitive blocks with non-zero weights and positive bond dimensions, the asymptotic overlap condition \(O_{A_kA_k}(N)\to 1\) is a derived consequence, and ordering the blocks by non-increasing weight modulus produces a normal canonical form.
If \((\mu _k,A_k)\) satisfies the normal canonical form conditions, then \(O_{A_kA_k}(N)\to 1\) for every block \(k\).
Fix a block \(k\). Irreducibility and the TP normalization give a nonzero positive semidefinite fixed point \(\rho \) of \(\mathcal{E}_{A_k}\) with fixed-point projector \(P\); primitivity gives the complementary gap \(\rho _{\operatorname{spec}}(\mathcal{E}_{A_k}-P){\lt}1\) (the corresponding theorem in the companion quantum-channel volume [ 216 , “Complementary transfer-map gap for primitive maps” ] ). Theorem 5.10.2 then yields \(O_{A_kA_k}(N)\to 1\).
Suppose \(A\) generates the same MPV family as \(\bigoplus _k\mu _kA_k\), and that each \(A_k\) is irreducible, satisfies \(\sum _i(A_k^i)^\dagger A_k^i=\mathbb {1}\), and has primitive transfer map, all \(\mu _k\neq 0\), and all bond dimensions are positive. Then there exist a blocking length \(p{\gt}0\), weights \((\nu _k)\), and blocked tensors \((B_k)\) such that \(A^{[p]}\) generates the same MPV family as \(\bigoplus _k\nu _kB_k\), and \((\nu _k,B_k)\) satisfies the normal canonical form conditions. The weight moduli need not be distinct; equal-modulus blocks are allowed.
The given blocks are already primitive, so take \(p=1\). Reorder the blocks by non-increasing weight modulus, with ties allowed, and set \(\nu _k:=\mu _k\) and \(B_k:=A_k\) in that order. Irreducibility, TP normalization, primitivity, nonzero weights, and positive bond dimensions are preserved, so the reordered family satisfies Definition 7.9.2 and generates the same MPV family as \(A\).
Theorem D.6.2 reorganizes an already reduced family of primitive, irreducible blocks into normal canonical form by sorting their weight moduli. The reduction from an arbitrary MPS tensor, including the structural bond-dimension bound, is established by Theorem 7.11.1.2 in Section 7.11.1 following [ 13 , Section 2.3 ] .