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 \(i\).

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 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.

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.

For a fixed period \(m\), a site-periodic tensor family is a family of local tensors indexed by \(\{ 0,\ldots ,m{-}1\} \), with local physical dimension \(d\) and bond dimension \(D\).

For site-periodic tensors \(A,B\) at fixed period \(m\):

• equality of the induced site-periodic states;

• cyclic gauge equivalence.

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 \(B\) (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 tensors \(A\) and \(B\) of possibly different bond dimensions \(D_1\) and \(D_2\), we write \(\mathcal{V}(A) = \mathcal{V}(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every word length \(N\), including the empty word. This algebraic relation is used for exact direct-sum identities.

For tensors \(A\) and \(B\) of possibly different bond dimensions, we write \(\mathcal{V}^+(A) = \mathcal{V}^+(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every \(N {\gt} 0\).

Let \(A\) and \(B\) have the same physical alphabet and respective bond dimensions \(D_A\) and \(D_B\). A reduction from \(B\) to \(A\) is a pair \(V\in \mathbb {C}^{D_A\times D_B}\) and \(W\in \mathbb {C}^{D_B\times D_A}\) such that

for every word \(\mathbf a\), including the empty word. This is the reduction of [ 221 , Proposition 20 and the following definition ] .

Two tensors \(A\) and \(B\) (of possibly different bond dimensions) generate nonzero proportional MPV families if for every \(N{\gt}0\) there is a nonzero scalar \(c_N \in \mathbb {C}\) such that \(|V^{(N)}(A)\rangle = c_N |V^{(N)}(B)\rangle \). This is the projective reading of the proportionality hypothesis in [ 13 , Theorem 2.10 ] .

Two tensors \(A\) and \(B\) are eventually nonzero proportional if, for all sufficiently large \(N\), there is a nonzero scalar \(c_N \in \mathbb {C}\) such that \(|V^{(N)}(A)\rangle = c_N |V^{(N)}(B)\rangle \). This auxiliary form is used with the eventual linear-independence statement Lemma E.1.7.

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 \(i\),

Two tensors are unit gauge-phase equivalent when the gauge-phase equation holds with \(|\zeta |=1\).

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

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

A tensor \(A\) is normal [ 17 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

Here, a block-injective canonical form for a tensor consists of:

1. a number of blocks \(r \ge 1\),

2. bond dimensions \(D_1, \ldots , D_r\),

3. injective tensors \(\{ A_k^i\} _{i=0}^{d-1}\) with \(A_k^i \in M_{D_k}(\mathbb {C})\) for each block \(k \in \{ 1,\ldots ,r\} \),

4. scaling factors \(\mu _1, \ldots , \mu _r \in \mathbb {C}\).

The associated tensor is the block-diagonal tensor

This block-injective formulation of the canonical form is used in [ 228 ] ; see also  [ 17 ] for the normal canonical form.

For a finite direct sum \(\bigoplus _k \mathbb {C}^{D_k}\), the block projection \(P_j\) is the matrix that is the identity on the \(j\)-th summand and zero on all other summands.

A matrix on \(\bigoplus _k \mathbb {C}^{D_k}\) is block diagonal if it is a direct sum of square matrices, one on each summand.

Given \(r\) blocks with bond dimensions \((D_k)_{k=1}^r\), per-block tensors \(\{ A_k^i\} \), and scaling factors \((\mu _k)_{k=1}^r\), the associated block-diagonal tensor is the tensor of total bond dimension \(D = \sum _{k=1}^r D_k\) defined by

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

We regard \(|V^{(N)}(A)\rangle \) as an element of \(\mathbb {C}^{d^N}\) indexed by configurations \(\sigma \in \{ 0,\ldots ,d{-}1\} ^N\), with \(\sigma \)-component \(V^{(N)}(A)_\sigma \) as in Definition 2.1.7. This allows us to use the standard Euclidean inner product on \(\mathbb {C}^{d^N}\).

The MPV inner product between two tensors \(A\) and \(B\) at system size \(N\) is

i.e. the Euclidean inner product on \(\mathbb {C}^{d^N}\) (conjugate-linear in the first argument).

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 2.8.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.8.2.