Dependencies

If \(D\ge 1\) and \(A\) is \(0\)-block injective, then \(D=1\).

Let \((A_x)_{x\in X}\) be a finite separated normal-canonical family whose blocks have positive virtual bond dimension. Then there is a positive integer \(L\) such that every blocked tensor \(A_x^{[L]}\) is one-site injective. The same conclusion holds simultaneously for two finite separated normal-canonical families whose blocks have positive virtual bond dimension.

Here the family is a basis of normal-canonical representatives in the sense of Definition 8.9.1.2; the blocks are distinct and the weight moduli are non-increasing but need not be strictly ordered. The representative family is assumed to have been selected; this statement does not reconstruct the full CPSV sector decomposition with repeated copies inside one sector. This scope restriction is recorded in [ con26g ] .

Under the hypotheses of Theorem 7.4.2, one has \(\iota (A)\le D^2-\operatorname{kr}(A)+1\), where \(\iota (A)=\min \{ n\ge 1:S_n(A)=M_{D}(\mathbb {C})\} \).

Let \(A\) be an MPS tensor with bond dimension \(D\ge 1\), satisfying \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and suppose that \(A\) is primitive in the sense of Definition 7.1.1.3. If some Kraus operator \(A^{i_0}\) is non-invertible and has an eigenvector \(\varphi \neq 0\) with nonzero eigenvalue \(\mu \), then

Under the hypotheses of Theorem 7.4.6, one has \(\iota (A)\le D^2\).

For a finite matrix family \(A\), let \(\operatorname{alg}(A)\) be the unital \(\mathbb {C}\)-subalgebra of \(M_{D}(\mathbb {C})\) generated by the matrices \(\{ A^i\} _{i=0}^{d-1}\): \(\operatorname{alg}(A)=\mathbb {C}\langle A^i:i=0,\ldots ,d-1\rangle \subseteq M_{D}(\mathbb {C})\).

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 \(A^{[L]}\).

The cumulative span at level \(n\) is \(T_n(A)=S_0(A)+S_1(A)+\cdots +S_n(A)\).

Given an MPS tensor \(A\) and a vector \(\varphi \in \mathbb {C}^D\), the cumulative vector span is

This is  [ SPGWC10 , proof of Lemma 2(a) ] .

Given a word \(w = (i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\), the word evaluation is the matrix product

The empty word evaluates to the identity, \(A^\varnothing = \mathbb {1}_D\). In tensor-network notation,

in which each black node denotes the same local tensor \(A\), the virtual legs remain open, and the physical legs are labelled by the word \((i_1,\ldots ,i_L)\).

The Kraus family has eventually full Kraus rank if there is a positive integer \(i\) such that \(S_i(A)=M_{D}(\mathbb {C})\).

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).

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

A subspace \(W\le \mathbb {C}^D\) is \(A\)-invariant if \(A^iW\subseteq W\) for every \(i\).

The matrices in \(A\) act irreducibly on \(\mathbb {C}^D\) if every \(A\)-invariant subspace is trivial, i.e. equal to \(0\) or to \(\mathbb {C}^D\).

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

A tensor \(A\) is \(L\)-block injective if

An injective tensor is \(1\)-block injective.

A (translation-invariant, PBC) MPS tensor with physical dimension \(d\) and bond dimension \(D\) is a collection of matrices \(\{ A^i\} _{i=0}^{d-1}\), where \(A^i \in M_{D}(\mathbb {C})\), indexed by a physical index \(i \in \{ 0, \ldots , d{-}1\} \). Such a tensor defines an MPV family. Diagrammatically,

The black node denotes the tensor \(A\), the horizontal legs are virtual, and the upper leg is the physical index \(i\).

For an MPS tensor \(A\), its fixed-length vector span is the fixed-length vector span of the underlying finite matrix family.

A tensor \(A\) is normal [ CPGSV21 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

Let \(A\) be an MPS tensor with Kraus family \(\{ A^i\} _{i=0}^{d-1}\), and let \(X\in M_{D}(\mathbb {C})\). The one-step augmentation of \(A\) by \(X\) is the tensor \(\widetilde A\) with physical dimension \(d+1\) whose first Kraus operator is \(X\) and whose remaining Kraus operators are those of \(A\):

The transfer map is primitive in the sense of [ SPGWC10 ] if there is a positive integer \(q\) such that

for every nonzero \(\varphi \in \mathbb {C}^D\). The same \(q\) works for all nonzero vectors.

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.

We call the tensor strongly irreducible here if there is a matrix \(\rho {\gt}0\) such that

and \(\mathcal{E}_A\) is irreducible. The condition in [ SPGWC10 , Proposition 3(c) ] requires that \(1\) be the only peripheral eigenvalue and that its eigenspace be one-dimensional, generated by a positive-definite fixed point. The definition above strengthens this condition by also requiring irreducibility explicitly.

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.

The full-Kraus-rank index is

If the set is empty, its value is defined to be zero.

The Kraus rank is \(\operatorname{kr}(A)=\dim S_1(A)\).

The primitivity index is

If the set is empty, its value is defined to be zero.

The word span at length \(n\) is the subspace

This is  [ SPGWC10 , Section II ] .

If the blocked tensor \(A^{[L]}\) satisfies \(S_n(A^{[L]})=M_{D}(\mathbb {C})\), then \(S_{nL}(A)=M_{D}(\mathbb {C})\). In particular, \(A\) is \(L\)-block injective if and only if \(A^{[L]}\) is one-site injective.

If \(A\) is a primitive MPS tensor with PSD fixed point \(\rho \), then \((\mathcal{E}_A-P_\rho )^n\to 0\) in operator norm.

For every \(n\), one has \(\dim T_n(A)\le D^2\).

For every \(n\), one has \(T_n(A)\le T_{n+1}(A)\).

Suppose \(A^{i_0}\varphi =\mu \varphi \) with \(\mu \neq 0\). Then \(K_n(A,\varphi )=H_n(A,\varphi )\) for every \(n\). In particular, if \(K_n(A,\varphi )=\mathbb {C}^D\), then \(H_n(A,\varphi )=\mathbb {C}^D\).

If \(A\) is a primitive MPS tensor with PSD fixed point \(\rho \), then every fixed point \(\sigma \) of \(\mathcal{E}_A\) is proportional to \(\rho \): \(\sigma =\frac{\operatorname{tr}(\sigma )}{\operatorname{tr}(\rho )}\rho \).

If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then the transfer map \(\mathcal{E}_A\) is irreducible.

The proof uses fixed-point uniqueness (Lemma C.5.5): every PSD fixed point is proportional to \(\rho \), so Wolf’s criterion for irreducibility (a positive-definite fixed point together with uniqueness of positive fixed points implies irreducibility) applies directly.

A tensor \(A\) is \(L\)-block injective if and only if its \(L\)-blocked tensor \(A^{[L]}\) is injective.

If the one-site matrices \(\{ A^i\} _{i=0}^{d-1}\) span \(M_{D}(\mathbb {C})\), then the length-one word products span \(M_{D}(\mathbb {C})\), so \(A\) is \(1\)-block injective.

Let \(X\in S_1(A)\), and let \(\widetilde A\) be the one-step augmentation of \(A\) by \(X\). If \(A\) is normal, then \(\widetilde A\) is normal, and \(\operatorname{kr}(\widetilde A)=\operatorname{kr}(A)\).

Let \(X\in S_1(A)\), and let \(\widetilde A\) be the one-step augmentation of \(A\) by \(X\). Then \(S_n(\widetilde A)=S_n(A)\) for every \(n\ge 0\).

If \(A\) is a primitive MPS tensor with PSD fixed point \(\rho \) and \(\rho \) is positive definite, then \(A\) is an irreducible tensor. This is part of  [ SPGWC10 , Proposition 3 ] (see also [ CPGSV16 , Section 2.3 ] ).

Let \(A\) be a normal MPS tensor with bond dimension \(D\ge 1\). Suppose that a Kraus operator \(A^{i_0}\) is non-invertible and that there exist a nonzero vector \(\varphi \in \mathbb {C}^D\) and a nonzero scalar \(\mu \) such that \(A^{i_0}\varphi =\mu \varphi \). Then, for every \(\psi \in \mathbb {C}^D\), \(|\varphi \rangle \! \langle \psi |\in S_{D^2-D+1}(A)\).

If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then it satisfies the strengthened condition in Definition 7.1.1.6: \(\mathcal{E}_A\) is irreducible, the fixed point \(\rho \) is positive definite, and the peripheral spectrum of \(\mathcal{E}_A\) is \(\{ 1\} \). The source clause is [ SPGWC10 , Proposition 3(c) ] ; the definition used here records irreducibility separately.

If \(T_N(A)=M_{D}(\mathbb {C})\) and \(\varphi \neq 0\), then \(K_N(A,\varphi )=\mathbb {C}^D\).

One has \(S_m(A)S_n(A)\subseteq S_{m+n}(A)\): the product of word spans at lengths \(m\) and \(n\) is contained in the word span at length \(m+n\).

One has \(S_L(A)=M_{D}(\mathbb {C})\) if and only if \(A\) is \(L\)-block injective.

If \(\operatorname{alg}(A)=M_{D}(\mathbb {C})\) and \(\mathbb {1}\in S_1(A)\), then \(A\) is normal.

Let \(A\) be a left-canonical normal MPS tensor with bond dimension \(D{\gt}0\). Then there is a positive blocking length \(L\le D^4\) such that the blocked tensor \(A^{[L]}\) is one-site injective.

This is the blocked-injectivity form of the quantum Wielandt input used in the Fundamental Theorem for translation-invariant tensors. It corresponds to the statement in [ CPGSV16 , Section II ] that every normal tensor becomes injective after at most \(D^4\) blockings, using the index estimate of  [ SPGWC10 , Theorem 1 ] . The left-canonical/trace-preserving hypothesis is included because this is the canonical-form context in which the blocked-injectivity input is used.

Assume \(D{\gt}0\). If \(A\) acts irreducibly on \(\mathbb {C}^D\), then the generated unital subalgebra is the full matrix algebra: \(\operatorname{alg}(A)=M_{D}(\mathbb {C})\). This is the complex finite-dimensional case of Burnside’s theorem (equivalently, Jacobson’s density theorem); see  [ Jac09 ] .

Let \((A_j)_{j=1}^g\) be a finite family of positive-bond-dimension, left-canonical normal tensors. There is a positive integer \(L\) such that every \(A_j\) is \(L\)-block injective.

If \(T_N(A)=M_{D}(\mathbb {C})\) and \(\mathbb {1}\in S_1(A)\), then \(A\) is normal.

Let \(D{\gt}0\). If \(A\) is normal, then \(T_{D^2}(A)=M_{D}(\mathbb {C})\).

Let \(D{\gt}0\). If \(A\) is normal, then \(T_{D^2-d'+1}(A)=M_{D}(\mathbb {C})\), where \(d'=\dim S_1(A)=\operatorname{kr}(A)\). This is the cumulative-span estimate used to prove the sharp trace conclusion in [ SPGWC10 , Lemma 1 ] .

If \(T_n(A)=T_{n+1}(A)\), then \(T_m(A)=T_n(A)\) for every \(m\ge n\).

If \(T_n(A)\neq T_{n+1}(A)\), then \(\dim T_{n+1}(A){\gt}\dim T_n(A)\).

Let \(A\) be a normal MPS tensor and let \(A^{i_0}\varphi =\mu \varphi \), where \(\mu \neq 0\) and \(\varphi \neq 0\). Then \(K_{D-1}(A,\varphi )=\mathbb {C}^D\). This cumulative form supplies the dimension-growth step in [ SPGWC10 , Lemma 2(a) ] .

Let \(D{\gt}0\), suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and assume that \(A\) is primitive in the sense of Definition 7.1.1.3. If \(\varphi \neq 0\), \(\mu \neq 0\), and \(A^{i_0}\varphi =\mu \varphi \), then \(H_{D-1}(A,\varphi )=\mathbb {C}^D\). This is  [ SPGWC10 , Lemma 2(a) ] .

Let \(A\) be a left-canonical normal MPS tensor with bond dimension \(D{\gt}0\). Then \(S_{D^4}(A)=M_{D}(\mathbb {C})\); equivalently, \(A\) is \(D^4\)-block injective.

Let \(A\) be a normalized MPS tensor, and suppose \(\mathcal{E}_A(X)=\mu X\) with \(X\neq 0\), \(\mu \neq 1\), and \(\mu ^p=1\). Then there is a nonzero Hermitian matrix \(H\) such that

Moreover, \(H\) is not positive semidefinite.

If \(A\) is irreducible as a tensor (Definition 8.1.1.1), then the Kraus operators act irreducibly on \(\mathbb {C}^D\) (Definition 7.7.3.3).

If \(A\) is normal and \(L{\gt}0\), then the blocked tensor \(A^{[L]}\) is also normal.

Assume:

1. \(H_n(A,\varphi )=\mathbb {C}^D\) (length-\(n\) word products applied to \(\varphi \) span all of \(\mathbb {C}^D\)), and

2. for each standard basis vector \(e_j\), the rank-one operator \(|\varphi \rangle \! \langle e_j|\) lies in \(S_m(A)\).

Then \(S_{n+m}(A)=M_{D}(\mathbb {C})\).

Suppose there is a length \(q\geq 0\) such that \(H_q(A,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \in \mathbb {C}^D\). Let \(p{\gt}0\). If \(\rho \) and \(\sigma \) are nonzero positive-semidefinite fixed points of \(\mathcal{E}_A^p\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

The conclusion of Theorem  [ LTC26 , “Eigenvector extraction from a full cumulative span” ] holds for the matrix family of an MPS tensor whenever its cumulative span at level \(N\) is the full matrix algebra.

Let \(D{\gt}0\), suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and assume that \(A\) is primitive in the sense of Definition 7.1.1.3. Then there is a word \(w\) with

This is the sharp trace conclusion of [ SPGWC10 , Lemma 1 ] .

Let \(D{\gt}0\). If \(A\) is normal, then there exists a word \(w\) of length at most \(D^2-\operatorname{kr}(A)+1\) such that \(\operatorname{tr}(A^w)\neq 0\). This normality variant has the same sharp length bound as [ SPGWC10 , Lemma 1 ] ; the source hypotheses are stated in Theorem 7.2.4.

If \(A\) is normal and \(D\ge 1\), then there exists a word \(w\) of positive length \(1\le |w|\le D^2-\operatorname{kr}(A)+1\) such that \(\operatorname{tr}(A^w)\neq 0\). This strengthens Theorem 7.2.5 by requiring \(|w|\ge 1\), a feature needed for the blocking argument in the general Wielandt bound below.

If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then \(A\) is normal (has eventually full Kraus rank). No identity-in-the-one-step-span condition is needed.

This connects the complementary-gap condition in Definition 7.6.1 directly to normality.

Let \(A\) be a left-canonical MPS tensor with \(D{\gt}0\), tensor-irreducible, and with peripheral spectrum \(\{ 1\} \). Tensor irreducibility implies irreducibility of the completely positive transfer map, so \(A\) is normal.

Under the hypotheses of Theorem 7.1.1.13, paper primitivity implies

Proposition 1 of  [ SPGWC10 ] states this inequality for every quantum channel; the present theorem records its specialization under normalization and paper primitivity.

Let \(D{\gt}0\), suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and assume that \(A\) is primitive in the sense of Definition 7.1.1.3. Suppose \(A^{i_0}\) is not invertible and \(A^{i_0}\varphi =\mu \varphi \) with \(\varphi \neq 0\) and \(\mu \neq 0\). Then, for every \(\psi \in \mathbb {C}^D\),

This is the quantitative rank-one conclusion of [ SPGWC10 , Lemma 2(b) ] .

Let \(A\) be a left-canonical MPS tensor with \(D{\gt}0\). Suppose that \(A\) is tensor-irreducible and that its transfer map is primitive. Then there exists a positive blocking length \(L\) such that \(A^{[L]}\) is one-site injective.

If \(\mathcal{E}_A(X)=\mu X\), then \(\mathcal{E}_A^n(X)=\mu ^nX\) for every \(n\geq 0\).

If \(K_n(A,\varphi )=K_{n+1}(A,\varphi )\), then \(K_m(A,\varphi )=K_n(A,\varphi )\) for every \(m\ge n\).

Let \(D{\gt}0\). If \(A\) is normal, then \(T_{D^2}(A)=M_{D}(\mathbb {C})\). This cumulative estimate supports the proof of [ SPGWC10 , Lemma 1 ] ; it is not a conclusion of source Theorem 1.

Let \(A\) be a normal MPS tensor with bond dimension \(D{\gt}0\), and suppose that some Kraus operator \(A^{i_0}\) is invertible. Then

Let \(A\) be an MPS tensor with bond dimension \(D\ge 1\), satisfying \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and suppose that \(A\) is primitive in the sense of Definition 7.1.1.3. If \(S_1(A)\) contains an invertible matrix \(X\), then

In the terminology of  [ SPGWC10 ] , this is the case where the first application space \(S_1(A)\) contains an invertible operator. This is the exact word-span form of [ SPGWC10 , Theorem 1, case (2) ] .

Let \(A\) be an MPS tensor with bond dimension \(D\ge 1\), satisfying \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and suppose that \(A\) is primitive in the sense of Definition 7.1.1.3. If \(S_1(A)\) contains a non-invertible matrix \(X\) with a nonzero eigenvalue, then \(S_{D^2}(A)=M_{D}(\mathbb {C})\). More explicitly, it is enough to have \(X\varphi =\mu \varphi \) with \(\varphi \neq 0\) and \(\mu \neq 0\). In the terminology of  [ SPGWC10 ] , this is the case where the first application space \(S_1(A)\) contains a non-invertible operator with a nonzero eigenvalue. This is the exact word-span form of [ SPGWC10 , Theorem 1, case (3) ] .

Let \(A\) be a normalized MPS tensor with \(D{\gt}0\), \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), that is primitive in the sense of Definition 7.1.1.3. Then

where \(\iota (A)=\min \{ n\ge 1:S_n(A)=M_{D}(\mathbb {C})\} \) is the full-Kraus-rank index. This is  [ SPGWC10 , Theorem 1, case (1) ] .

Let \(D{\gt}0\) and suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). The following are equivalent:

1. there is a positive \(q\) for which \(H_q(A,\varphi )=\mathbb {C}^D\) for every \(\varphi \neq 0\);

2. there is a positive \(i\) for which \(S_i(A)=M_{D}(\mathbb {C})\);

3. \(\mathcal{E}_A\) is strongly irreducible.

Proposition 3 of  [ SPGWC10 ] states the same equivalence with item (3) replaced by the literal source condition: \(1\) is the only peripheral eigenvalue, and its one-dimensional eigenspace is generated by a positive-definite fixed point. Here item (3) includes the additional irreducibility condition.

Let \(D{\gt}0\) and suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). If there is a positive \(n\) for which \(S_n(A)=M_{D}(\mathbb {C})\), then there is a unique positive definite density matrix \(\rho \in M_{D}(\mathbb {C})\) such that every fixed point of \(\mathcal{E}_A\) is a scalar multiple of \(\rho \).

This is  [ Wol12 , Theorem 6.15 ] .

Under normalization, primitivity in Wolf’s sense is equivalent to the conjunction of eventual full Kraus rank, normality, and strong irreducibility.

Under normalization, primitivity in Wolf’s sense is equivalent to eventual full Kraus rank.

For every \(n\),

Consequently, if \(L{\gt}0\) and \(A\) is \(L\)-block injective, then \(A\) is \(m\)-block injective for every \(m\ge L\). This auxiliary length-shift statement combines the block-injectivity discussion in  [ CPGSV16 , Section II ] with [ CPGSV16 , Appendix C.3, Lemma L ] ; it is not stated there as a separate lemma.

Suppose

and \(S_L(A)=M_{D}(\mathbb {C})\). Then \(S_m(A)=M_{D}(\mathbb {C})\) for every \(m\ge L\). Equivalently, \(L\)-block injectivity propagates to every larger homogeneous length.