Dependencies

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

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 invertible, then

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 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 \(i\).

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.

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 \(A\):

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.

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

This is  [ SPGWC10 , Section II ] .

If the generated unital algebra satisfies \(\operatorname{alg}(A)=M_{D}(\mathbb {C})\), then there exists \(N\) such that \(T_N(A)=M_{D}(\mathbb {C})\).

If \(A\) is \(L\)-block injective and \(X\neq 0\), then

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 \(\dim T_n(A)\le D^2\).

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 the letters of \(A\) act irreducibly on \(\mathbb {C}^D\), then \(A\) has no nontrivial invariant orthogonal projection.

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.

Every Kraus operator \(A^i\) belongs to the one-step span \(S_1(A)\).

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, then for every \(n\ge 0\) there exists a word \(w\) of length exactly \(n\) with \(A^w\neq 0\).

If \(A\) is a primitive MPS tensor, then \(\mathcal{E}_A^n\neq 0\) for all \(n\ge 0\).

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

One has \(S_{nL}(A)\subseteq S_n(A^{[L]})\) for all \(n,L\).

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 \(D\ge 2\), then \(S_0(A)\neq M_{D}(\mathbb {C})\).

If \(A^{w_0}\varphi =\mu \varphi \) for a word \(w_0\) of length \(L\), then \((A^{[L]})^{j_0}\varphi =\mu \varphi \), where \(j_0\) is the encoded blocked index corresponding to \(w_0\).

If \(\mathbb {1}\in S_1(A)\), then \(T_n(A)=S_n(A)\) for every \(n\).

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 \(T_{D^2}(A)=M_{D}(\mathbb {C})\).

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 \(D{\gt}0\). If \(A\) is normal, then there exist a word \(w_0\) with \(|w_0|\le D^2\), a nonzero scalar \(\mu \neq 0\), and a nonzero vector \(\varphi \neq 0\) such that \(A^{w_0}\varphi =\mu \varphi \).

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) ] .

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

Let \(D{\gt}0\). If \(A\) is normal, then there exists a word \(w\) of length at most \(D^2\) such that \(\operatorname{tr}(A^w)\neq 0\). This is the weaker \(D^2\) corollary of [ SPGWC10 , Lemma 1 ] . The sharp source statement is Theorem 7.2.4.

Let \(A\) have positive bond dimension and satisfy \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). If \(A\) is tensor-irreducible and the transfer map is primitive, then \(O_{AA}(N)\to 1\).

If \(A\) is a primitive MPS tensor, then \(O_{AA}(N)\to 1\).

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 \(N\to \infty \).

Suppose \(A\) has bond dimension \(D{\gt}0\), is normal, and \(N_0\ge 1\) with \(S_{N_0}(A)=M_{D}(\mathbb {C})\). Then there exist words \(\sigma _0,\tau _0\) of length \(N_0\), nonzero vectors \(\varphi ,\psi \), nonzero scalars \(\mu ,\nu \), and \(m\in \mathbb {N}\) such that:

1. \(A^{\sigma _0}\varphi =\mu \varphi \),

2. \((A^{\tau _0})^T\psi =\nu \psi \),

3. \(\varphi \psi ^T\in S_m(A^{[N_0]})\).

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

Suppose \(A\) has bond dimension \(D{\gt}0\), is normal, and \(L{\gt}0\), and we have:

1. a word \(w_0\) of length \(L\) with eigenvector \(A^{w_0}\varphi =\mu \varphi \) (\(\mu \neq 0\), \(\varphi \neq 0\)),

2. a word \(\tau _0\) of length \(L\) with transpose eigenvector \((A^{\tau _0})^T\psi =\nu \psi \) (\(\nu \neq 0\), \(\psi \neq 0\)),

3. a rank-one element \(\varphi \psi ^T\in S_m(A^{[L]})\) for some \(m\).

Then there exists \(N\) such that \(S_N(A)=M_{D}(\mathbb {C})\).

Remark. The proof gives the explicit bound \(N=((D{-}1)+m+(D{-}1))L\); the theorem states only the existential conclusion.

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) ] .

If \(A\) is a normal MPS tensor with bond dimension \(D{\gt}0\), then there exists \(N\) such that \(S_N(A)=M_{D}(\mathbb {C})\). This is a blocked fixed-length construction under the stated normality hypothesis. The quantitative result of [ SPGWC10 , Lemma 2(b) ] is Theorem 7.4.5.

If \(\mathbb {1}\in S_1(A)\), then \(S_n(A)\le S_{n+1}(A)\) for every \(n\).