Dependencies

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 \((i_1,\ldots ,i_L)\).

Two tensors \(A\) and \(B\) of the same bond dimension \(D\) are gauge equivalent if there exists an invertible matrix \(X \in \mathrm{GL}_D(\mathbb {C})\) such that, for every \(i \in \{ 0,\ldots ,d{-}1\} \),

In tensor-network notation, (8) is represented by

where the black node denotes the tensor \(A\), the upper leg is the physical index \(i\), and the two red side nodes denote the gauge matrices acting on the virtual legs.

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: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

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 matrix product vector (MPV) of a tensor \(A\) at system size \(N\) is the vector

Equivalently, for a configuration \(\sigma = (i_1, \ldots , i_N) \in \{ 0,\ldots ,d{-}1\} ^N\), we write

The coefficient function \(\sigma \mapsto V^{(N)}(A)_\sigma \) gives the components of the vector in (2); the displayed ket is the corresponding vector in \((\mathbb {C}^d)^{\otimes N}\). For a general word \(w\), we also write \(c_w(A) := \operatorname{tr}(A^w)\). The MPV family generated by \(A\) is the collection \(\mathcal{V}(A) = \bigl\{ |V^{(N)}(A)\rangle \bigr\} _{N \ge 1}\). The coefficient \(V^{(N)}(A)_\sigma \) is the periodic contraction

of \(N\) copies of the local tensor \(A\), with the outer virtual legs closed by the trace.

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 a tensor \(A\), the trace pairing map \(\Phi _A : M_{D}(\mathbb {C}) \to \mathbb {C}^d\) is defined by

Let \(\Phi , \Psi : V \to W\) be \(\mathbb {C}\)-linear maps with \(V\) finite-dimensional. If \(\ker \Phi = \{ 0\} \) and \(\operatorname{range}\Phi \subseteq \operatorname{range}\Psi \), then \(\ker \Psi = \{ 0\} \).

If \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is a surjective multiplicative \(\mathbb {C}\)-linear map, then \(T(\mathbb {1}) = \mathbb {1}\), so \(T\) is a \(\mathbb {C}\)-algebra homomorphism.

If \(A\) and \(B\) generate the same matrix-product vectors, then \(\operatorname{tr}(A^w) = \operatorname{tr}(B^w)\) for every word \(w\).

Let \(D \ge 1\). If \(A\) is injective and \(A\), \(B\) generate the same matrix-product vectors, then the \(B^i\) are not all zero.

Let \(B\) be an MPS tensor of bond dimension \(D_B\), and let \(\{ M_\gamma \} _{\gamma \in \Gamma }\) be a finite family of normal MPS tensors on the same physical alphabet, with \(D_\gamma {\gt}0\). For distinct \(\gamma ,\delta \) of equal bond dimension, assume there are no \(X\in \mathrm{GL}(D_\gamma ,\mathbb {C})\) and \(z\in \mathbb {C}\setminus \{ 0\} \) such that \(M_\delta ^i=zXM_\gamma ^iX^{-1}\) for every physical index \(i\). Suppose that for every \(N{\gt}0\) there are coefficients \(c_\gamma ^{(N)}\) satisfying

Then there are integers \(n_\gamma \geq 0\) and nonzero complex weights \(\mu _{\gamma k}\), indexed by \(0\leq k{\lt}n_\gamma \), such that for every \(N{\gt}0\),

The multiplicities satisfy \(\sum _{\gamma \in \Gamma }n_\gamma D_\gamma \leq D_B\). There is a positive threshold \(N_0\) such that, for every \(N\geq N_0\), any coefficients satisfying (8) obey \(c_\gamma ^{(N)}=\sum _{k=0}^{n_\gamma -1}\mu _{\gamma k}^{N}\). The weight multisets are unique: any finite families of nonzero weights giving an expansion of the form (9) at all sufficiently large lengths have, for each \(\gamma \), exactly the same entries with the same multiplicities. Empty weight multisets are allowed.

Let \(A\) be an injective MPS tensor and \(B\) a tensor of the same bond dimension. If \(A\) and \(B\) generate the same matrix-product vectors, then there is \(X \in \mathrm{GL}_D(\mathbb {C})\) with \(B^i = X A^i X^{-1}\) for all \(i\). In tensor-network notation the gauge conjugates the local tensor on its virtual legs, with the physical index \(i\) on the upper leg:

Let \(A\) be an injective MPS tensor with \(D\ge 1\) and \(B\) a tensor of the same bond dimension. Suppose that for every \(N\ge 1\) there is \(c_N\in \mathbb {C}\) with \(|c_N|=1\) and \(|V^{(N)}(B)\rangle =c_N|V^{(N)}(A)\rangle \). Then there are \(\lambda \in \mathbb {C}\) with \(|\lambda |=1\) and \(X\in \mathrm{GL}_D(\mathbb {C})\) with \(B^i=\lambda \, XA^iX^{-1}\) for all \(i\), and \(c_N=\lambda ^N\). This is the single-block case of the Fundamental Theorem of matrix product vectors [ 13 , Theorem 2.10 ] , with unimodular proportionality constants.

If \(A\) is injective and \(A\), \(B\) generate the same matrix-product vectors, then there is a unique linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) with \(T(A^i) = B^i\) for all \(i\).

Under the hypotheses of Theorem 3.1.5, the map \(T\) satisfies \(T(MN) = T(M) T(N)\) for all \(M, N \in M_{D}(\mathbb {C})\).

Let \(\mu _1,\ldots ,\mu _n\) be nonzero complex numbers with \(\left|\sum _k\mu _k^L\right|=1\) for every \(L\geq 1\). Then \(n=1\) and \(|\mu _1|=1\).

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

If \(A\) is injective, then \(\ker \Phi _A = \{ 0\} \).