Dependencies

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 gauge-phase equivalent if there exist \(X \in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\setminus \{ 0\} \) such that, for every physical index \(i\),

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

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

An MPS tensor \(A=(A^i)_i\) is left-canonical when

Equivalently, its Kraus map is trace-preserving.

The mixed transfer operator (or cross transfer operator) for two MPS tensors \(A\) and \(B\) of the same bond dimension \(D\) is the linear map \(F_{AB}:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) defined by

For MPS tensors \(A\) of bond dimension \(D_1\) and \(B\) of bond dimension \(D_2\), possibly with \(D_1\neq D_2\), the rectangular mixed transfer operator is the map \(F_{AB}^{\mathrm{rect}}: M_{D_1\times D_2}(\mathbb {C})\to M_{D_1\times D_2}(\mathbb {C})\) defined by

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.

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.

Let \(a\) be an element of a complex Banach algebra with \(a^2=a\) and \(\rho _{\operatorname{spec}}(a){\lt}1\). Then \(a=0\).

Let \(P_0,\ldots ,P_{m-1}\) be orthogonal projections summing to the identity and cyclically permuted by a Kraus map, \(\sum _vK_vP_{k+1}K_v^\dagger =P_k\). Then the projections are mutually orthogonal, none of them vanishes, and each Kraus operator intertwines consecutive projections:

Let \(A\) and \(B\) have the same bond dimension. If \(\| \langle V^{(N)}(A) | V^{(N)}(B) \rangle \| \to 1\), then the mixed transfer map has spectral radius at least \(1\).

Let \(A\) be a left-canonical irreducible tensor whose adjoint transfer map has peripheral spectrum \(\{ \gamma ^j:0\le j{\lt}m\} \) for a primitive \(m\)th root of unity \(\gamma \), with blocked cyclic-sector decomposition \((C_u)_{u=1}^m\), \((P_u)_{u=1}^m\), and linear isomorphisms \(\varphi _u:M_{\dim C_u}(\mathbb {C})\xrightarrow {\sim }P_uM_{D}(\mathbb {C})P_u\). Assume explicitly that the \(P_u\) are orthogonal projections summing to \(\mathbb {1}\) and satisfying \(\mathcal{E}_{A^\dagger }(P_{u+1})=P_u\), that each sector has nonzero bond dimension, and that the compression maps satisfy

Here \(P_uA^{[m]}\) denotes the blocked tensor with letters \(P_u(A^{[m]})^\alpha \). Then every sector tensor satisfies \(\sigma _\partial (\mathcal{E}_{C_u})=\{ 1\} \) and is tensor-irreducible.

Let \(A\) be an irreducible left-canonical tensor of positive bond dimension. If \(O_{AA}(N)\to 1\), then \(A\) is normal.

Let \(A\) be an irreducible left-canonical tensor of positive bond dimension. If \(O_{AA}(N)\longrightarrow 1\), then the transfer map of \(A\) is primitive and \(A\) is normal.

Let \(A\) be a left-canonical irreducible tensor, and suppose the adjoint transfer map \(\mathcal{E}_{A^\dagger }\) has a positive-definite fixed point, is irreducible as a positive map, and has peripheral spectrum

where \(\gamma \) is a primitive \(m\)th root of unity and \(m\ge 1\). Then the spectral cyclic-sector decomposition of \(A^{[m]}\) has compression maps which may be chosen with rectangular support isometries \(V_k\) satisfying

These isometries also realize the blocked corner letters \(\varphi _k(C_k^\alpha )=P_k(A^{[m]})^\alpha P_k\).

Let \(A\) and \(B\) be normalized MPS tensors. Every eigenvalue \(\mu \) of \(F_{AB}\) satisfies \(|\mu |\leq 1\).

Let \(A\) and \(B\) be normalized MPS tensors of bond dimensions \(D_1\geq 1\) and \(D_2\geq 1\). Every eigenvalue \(\mu \) of \(F_{AB}^{\mathrm{rect}}\) satisfies \(|\mu |\leq 1\).

Under the hypotheses of Lemma 6.8.1.1, a product of \(\ell \) Kraus operators moves each cyclic projection back by \(\ell \) steps. In particular, every product of \(m\) Kraus operators commutes with every cyclic projection.

Let \(A\) be a normalized tensor of bond dimension \(D_1\) and \(B\) a normalized tensor of bond dimension \(D_2\). Suppose \(\rho _A,\rho _B\) are fixed points of the respective transfer maps and \(S_A,S_B\) are invertible matrices with \(S_AS_A^\dagger =\rho _A\) and \(S_BS_B^\dagger =\rho _B\) (for instance the positive definite Perron–Frobenius fixed points of Theorem 5.3.2 and their square roots, as in the right-canonical gauge of Theorem  [ LTC26 , “Right-canonical (unital) gauge” ] ). Gauge to unital families \(A'^i=S_A^{-1}A^iS_A\) and \(B'^i=S_B^{-1}B^iS_B\). If \(X\in M_{D_1\times D_2}(\mathbb {C})\setminus \{ 0\} \) satisfies \(F_{AB}^{\mathrm{rect}}(X)=\mu X\) with \(|\mu |=1\), then \(X'=S_A^{-1}X(S_B^\dagger )^{-1}\) is nonzero, the families \(A'\) and \(B'\) are unital, and, for every \(i\),

Let \(A\) and \(B\) be tensors of the same bond dimension. Suppose invertible gauges take them to \(A'\) and \(B'\), and suppose there are an invertible matrix \(X'\) and a scalar \(\mu \) with \(|\mu |=1\) such that \(A'^iX'=\mu X'B'^i\) for every physical index \(i\). Then \(A\) and \(B\) are gauge-phase equivalent.

Let \(a\) be a nonzero element of a complex normed algebra. If \(a^2=a\), then

For every \(X\in M_{D}(\mathbb {C})\) and \(N\geq 0\),

where \(A^\sigma =A^{\sigma _1}\cdots A^{\sigma _N}\) denotes the word evaluation of Definition 2.1.2.

Let \(A\) and \(B\) be injective normalized MPS tensors. If \(\rho _{\operatorname{spec}}(F_{AB})\geq 1\), then \(A\) and \(B\) are gauge-phase equivalent.

Let \(A\) and \(B\) be irreducible trace-preserving normalized MPS tensors of bond dimension \(D\). If \(\rho _{\operatorname{spec}}(F_{AB})\geq 1\), then \(A\) and \(B\) are gauge-phase equivalent.

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.

Let \(A\) and \(B\) be irreducible MPS tensors of the same bond dimension with \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\) and \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\). If \(\| \langle V^{(N)}(A) | V^{(N)}(B) \rangle \| \to 1\), then \(A\) and \(B\) are gauge-phase equivalent.

Let \(A\) and \(B\) be injective normalized tensors of the same bond dimension that are not gauge-phase equivalent. Then \(O_{AB}(N)\to 0\) as \(N\to \infty \).

Let \(A\) and \(B\) be irreducible trace-preserving normalized tensors of the same bond dimension that are not gauge-phase equivalent. Then \(O_{AB}(N)\to 0\) as \(N\to \infty \).

If \(D_1\neq D_2\) and both tensors are injective and normalized, then \(O_{AB}(N)\to 0\) as \(N\to \infty \).

If \(D_1\neq D_2\) and both tensors are irreducible and trace-preserving normalized, then \(O_{AB}(N)\to 0\) as \(N\to \infty \).

For tensors \(A\) and \(B\) of bond dimension \(D\geq 1\),

For tensors \(A\) of bond dimension \(D_1\geq 1\) and \(B\) of bond dimension \(D_2\geq 1\),

Let \(D{\gt}0\) and \(m{\gt}0\). Let \(A\) be a left-canonical irreducible MPS tensor with bond dimension \(D\), and suppose that the peripheral eigenvalues of its transfer map are exactly \(\{ z\in \mathbb {C}:z^m=1\} \). Then \(O_{AA}(mn)\longrightarrow m\).

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 \(N\to \infty \).

Assume \(D \ge 1\). Let \(A\) be an injective MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then the transfer map \(\mathcal{E}_A\) has a unique positive semidefinite fixed point \(\rho \) up to scaling, and \(\rho \) is positive definite.

Let \(A\) and \(B\) be injective tensors of bond dimensions \(D_1\) and \(D_2\). If there are \(X\in M_{D_1\times D_2}(\mathbb {C})\setminus \{ 0\} \) and \(\mu \in \mathbb {C}\) such that, for every physical index \(i\),

then \(D_1=D_2\).

Let \(A\) and \(B\) have bond dimensions \(D_1\) and \(D_2\), respectively. If \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\), then, for every \(X\in M_{D_1\times D_2}(\mathbb {C})\),

This is the decay mechanism for the off-diagonal transfer operators in [ CPGSV16 , Appendix B, equations (B.3)–(B.4) and the paragraph following equation (B.6) ] .

A nonzero multiplicative \(\mathbb {C}\)-linear endomorphism of \(M_{D}(\mathbb {C})\) is bijective.

For normalized tensors, \(\rho _{\operatorname{spec}}(F_{AB})\leq 1\).

Let \(a\) belong to a complex Banach algebra and let \(c\in \mathbb {C}\setminus \{ 0\} \). Then

Let \(E\) be trace-preserving with fixed point \(\rho \), where \(\operatorname{tr}(\rho )\neq 0\), and let \(P\) be the fixed-point projection of the corresponding definition in the companion quantum-channel volume  [ LTC26 , “Fixed-point projection” ] . If \(\rho _{\operatorname{spec}}(E-P){\lt}1\), then \(\operatorname{Tr}(E^n)\to 1\) as \(n\to \infty \).

Let \(A\) and \(B\) be injective normalized MPS tensors that are not gauge-phase equivalent. Then \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\).

Let \(A\) and \(B\) be irreducible trace-preserving normalized MPS tensors of the same bond dimension that are not gauge-phase equivalent. Then \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\).

Let \(A\) be an injective normalized tensor of bond dimension \(D_1\) and \(B\) an injective normalized tensor of bond dimension \(D_2\), with \(D_1\neq D_2\). Then \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\).

Let \(A\) and \(B\) be irreducible trace-preserving normalized tensors of bond dimensions \(D_1\) and \(D_2\), respectively, with \(D_1\neq D_2\). Then \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\).

Let \(A\) and \(B\) be injective normalized tensors that are not gauge-phase equivalent. Then \(F_{AB}^n(X)\to 0\) for every \(X\in M_{D}(\mathbb {C})\).

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.