Dependencies

Given \(r\) blocks with bond dimensions \((D_k)_{k=1}^r\), per-block tensors \(\{ A_k^i\} \), and scaling factors \((\mu _k)_{k=1}^r\), the associated block-diagonal tensor is the tensor of total bond dimension \(D = \sum _{k=1}^r D_k\) defined by

Given a tensor \(A\) and a blocking length \(L\), the \(L\)-blocked tensor is the tensor with physical index set \(\{ 0,\ldots ,d{-}1\} ^L\) (hence physical dimension \(d^L\)) and the same bond dimension \(D\), whose matrices are indexed by words \((i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\):

Blocking coarse-grains \(L\) neighbouring sites into one tensor: the inner virtual bonds are contracted, the \(L\) physical legs merge into a single composite index, and the two outer bonds remain as the bond of \(A^{[L]}\).

Two tensors \(A\) and \(B\) are eventually nonzero proportional if, for all sufficiently large \(N\), there is a nonzero scalar \(c_N \in \mathbb {C}\) such that \(|V^{(N)}(A)\rangle = c_N |V^{(N)}(B)\rangle \). This auxiliary form is used with the eventual linear-independence statement Lemma E.1.10.

An MPS tensor \(A\) has an invariant projection if there exists an orthogonal projection \(P\) with \(P\neq 0\), \(P\neq \mathbb {1}\), and \((\mathbb {1}-P)A^iP=0\) for all \(i\). A tensor is irreducible if it has no nontrivial invariant projection (Definition 8.1.1.1).

An MPS tensor \(A\) has invariant-projector closure if every orthogonal projection \(P\) satisfying \(PA^i=PA^iP\) for all \(i\) also satisfies \(A^iP=PA^iP\) for all \(i\). This is the projector hypothesis in the sufficient condition for canonical form of  [ CPGSV16 , Section 2 ] .

An MPS tensor \(A\) is irreducible if it has no nontrivial invariant projection.

Scaling factors \((\mu _k)_{k=1}^r\) and block tensors \((A_k)_{k=1}^r\) satisfy the normal canonical form conditions if:

1. each \(A_k\) is irreducible,

2. each \(A_k\) satisfies \(\sum _i(A_k^i)^\dagger A_k^i=\mathbb {1}\),

3. each \(\mathcal{E}_{A_k}\) is primitive (equivalently, \(\sigma _\partial (\mathcal{E}_{A_k})=\{ 1\} \)),

4. \(|\mu _1|\ge |\mu _2|\ge \cdots \ge |\mu _r|\),

5. all \(\mu _k\neq 0\),

6. all bond dimensions are positive.

Here “normal” is the spectral sense of  [ CPGSV16 ] , which by  [ SPGWC10 , Proposition 3 ] is equivalent to the eventual block-injectivity formulation of Definition 2.4.4.

The MPV overlap between two tensors \(A\) (of bond dimension \(D_1\)) and \(B\) (of bond dimension \(D_2\)) at system size \(N\) is

Diagrammatically, the overlap contracts the MPV ring of \(A\) against the conjugate ring of \(B\) along their shared physical indices, both virtual rings closed by the trace:

This pairing is linear in the first MPV coefficient and conjugate-linear in the second. By Lemma 2.7.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.7.2.

An MPS tensor \(A\) has no nontrivial \(p\)-periodic vectors if, for every isometry \(V\) with \(V^\dagger V=\mathbb {1}\) and \(A^iV=VB^i\) for all \(i\), with \(B\) irreducible (equivalently, for every restriction \(B\) of \(A\) to a minimal invariant subspace), and for every \(\rho {\gt}0\) and \(r{\gt}0\) with \(\mathcal{E}_B(\rho )=r\rho \), every eigenvalue of \(\mathcal{E}_B\) of modulus \(r\) equals \(r\). The eigenvalue \(r\) is the spectral radius of \(\mathcal{E}_B\) (Theorem  [ LTC26 , “Spectral radius is the Perron eigenvalue” ] ), so the condition states that no block of the invariant-subspace decomposition of  [ CPGSV16 , Section 2 ] , rescaled to spectral radius one, has a peripheral eigenvalue \(e^{2\pi iq/p}\neq 1\). Eigenvectors with such eigenvalues are the \(p\)-periodic vectors of  [ CPGSV16 , Section 2 ] .

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of  [ CPGSV16 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

\(A\) has primitive irreducible cyclic sectors of period \(m\ge 1\) if there is a family \((C_k)_{k=1}^m\) with \(A^{[m]}\sim \bigoplus _{k=1}^mC_k\) with unit weights, each \(C_k\) left-canonical, with primitive transfer map, tensor-irreducible, and of positive bond dimension.

Assume \(D{\gt}0\). A tensor \(A\) together with a matrix \(\rho \in M_{D}(\mathbb {C})\) is primitive if

where \(P\) is the fixed-point projection associated to \(\rho \). When the choice of \(\rho \) is irrelevant, we simply say that \(A\) is a primitive MPS tensor. This condition combines a complementary spectral gap with a nonzero positive semidefinite fixed point. It is not the paper definition in Definition 7.1.1.3, which is the uniform spreading condition (1). With the additional hypothesis \(\rho {\gt}0\), the complementary-gap condition implies strong irreducibility and hence paper primitivity by Theorem 7.1.1.13.

Two tensors \(A\) and \(B\) (of the same bond dimension) generate the same MPV family if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every system size \(N\).

For tensors \(A\) and \(B\) of possibly different bond dimensions \(D_1\) and \(D_2\), we write \(\mathcal{V}(A) = \mathcal{V}(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every word length \(N\), including the empty word. This algebraic relation is used for exact direct-sum identities.

For tensors \(A\) and \(B\) of possibly different bond dimensions, we write \(\mathcal{V}^+(A) = \mathcal{V}^+(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every \(N {\gt} 0\).

The transfer map associated to a tensor \(A\) is the linear map \(\mathcal{E}_A : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) defined by

Diagrammatically, the transfer map is the double-layer contraction

in which the upper node denotes \(A\), the lower node denotes \(A^\dagger \), and the physical index is summed over between them.

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.

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.

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.

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 [ CPGSV16 , Section 2.3 ] arise from the transfer eigenvalues of modulus \(r\).

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.

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 \(\mathcal{E}_B(\rho )=r\rho \).

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 \(E_kE_k^\dagger \).

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 \(\mathcal{V}(A)=\mathcal{V}(\bigoplus _{k=1}^rB_k)\).

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

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.

Suppose that isometries \(V_k\) satisfy

Then \(A\) and \(\bigoplus _k\mu _kB_k\) have the same matrix product vectors at every positive length.

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.

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 \(\mathcal{E}_A\).

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 \(\varphi _k(C_k^i)=P_kA^iP_k\).

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.

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 \(\alpha \).

Let \(A\) be an irreducible MPS tensor with \(D{\gt}0\) and \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). Then there exists a blocking length \(p{\gt}0\) such that \(\sigma _\partial (\mathcal{E}_{A^{[p]}})=\{ 1\} \).

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.

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

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.

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.

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 \(d^{[m_k]}\).

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 \(k\).

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 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 \(\varphi _k(C_k^i)=P_kA^iP_k\).

Let \(A\) be a left-canonical irreducible tensor. Then there is a period \(m\ge 1\) and a unit-weight decomposition of \(A^{[m]}\) into sector blocks \((C_k)_{k=1}^m\), each of which is left-canonical, has primitive transfer map, is tensor-irreducible, and has nonzero bond dimension. Equivalently, the cyclic-sector decomposition of \(A^{[m]}\) already satisfies the trace-preservation, primitivity, and irreducibility hypotheses needed for the common-blocking step (Lemma D.5.5).

If two tensors have eventually nonzero proportional MPV coefficients, then so do their blocked tensors after any positive blocking length.

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 \(p\).

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.

If \(A\) has invariant-projector closure and admits a nontrivial invariant projection, then it admits a nontrivial orthogonal projection commuting with every \(A^i\).

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 \(B_k\).

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 \(i\).

Every MPS tensor \(A\) admits a block-diagonal tensor \(\bigoplus _{k=1}^r A_k\) with each \(A_k\) irreducible, \(\mathcal{V}(A)=\mathcal{V}(\bigoplus _k A_k)\), and \(\sum _{k=1}^rD_k=D\).

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 [ CPGSV16 , Section 2.3 ] applied to an irreducible corner.

For any MPS tensor \(A\) and \(m,n\ge 0\),

If a finite matrix family satisfies

then every blocked family satisfies the same identity.

Suppose \(P\) is an orthogonal projection satisfying \((\mathbb {1}-P)A^iP=0\) for every physical index \(i\). Then there exist \(n,m\) with \(n+m=D\) and MPS tensors \(A_1,A_2\) of bond dimensions \(n\) and \(m\) such that \(A\) and the two-block tensor \(A_1\oplus A_2\) generate the same MPV family in the sense of Definition 2.3.3.

Under the hypotheses of Theorem 8.1.2, 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 2.3.3.

If \((\mu _k,A_k)\) satisfies the normal canonical form conditions, then \(O_{A_kA_k}(N)\to 1\) for every block \(k\).

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, \(\sum _{k=1}^rD_k\le D\).

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.

Let \(A\) be a normalized MPS tensor with a nonzero PSD fixed point \(\rho \) of the transfer map. If \(\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1\), where \(P\) is the fixed-point projection, then \(O_{AA}(N)\to 1\) as \(N\to \infty \).

If two tensors have the same MPV coefficients at every positive length, then so do their blocked tensors after any positive blocking length.

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 \(U\) is a unitary matrix, then \(A\) and \(U^\dagger A U\) generate the same MPV family.

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, [ Wol12 , Theorem 6.6 ] .

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 \(\sigma =(\sigma _1,\ldots ,\sigma _N)\),

For any MPS tensor \(A\), there exist nontrivial blocks \((B_k)_{k=1}^r\) with nonzero weights \((\mu _k)_{k=1}^r\), each block irreducible, left-canonical \((\sum _i(B_k^i)^\dagger B_k^i=\mathbb {1})\), and of positive bond dimension, such that, for every \(N\ge 1\) and every \(\sigma \),

and \(\sum _{k=1}^rD_k\le D\).

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.

If \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), then the transfer map \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) is a quantum channel (CPTP).

Let \(A\) be an MPS tensor and let \(c\in \mathbb {C}\). If

then, for every integer \(L\geq 0\),

Let \(A\) be an MPS tensor whose transfer map \(\mathcal{E}_A\) is irreducible. If \(\rho \geq 0\), \(\rho \neq 0\), and \(\mathcal{E}_A(\rho )=\rho \), then \(\rho \) is positive definite.

Let \(\rho \) be positive definite with \(\mathcal{E}_A(\rho )=r\rho \) for some \(r{\gt}0\). For \(B^i:=r^{-1/2}\rho ^{-1/2}A^i\rho ^{1/2}\),

If \(r=1\), then the unscaled gauge \(B^i:=\rho ^{-1/2}A^i\rho ^{1/2}\) is unital and generates the same MPV family as \(A\). This is the spectral-radius normalization and full-rank fixed-point gauge of [ PGVWC07 , Theorem 4 ] .