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, (5) 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.

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 \(A\) 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:

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 \(M, N \in M_{D}(\mathbb {C})\).

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

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