Dependencies

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

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.

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

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 [ PGVWC07 ] ; see also  [ CPGSV21 ] for the normal canonical form.

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 \(B\) 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.1.10.

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.

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 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 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 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 2.7.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.7.2.

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 \(\mathbb {C}^{d^N}\).

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 [ CPGSV16 , Theorem 2.10 ] .

A tensor \(A\) is normal [ CPGSV21 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

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) 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 \(N {\gt} 0\).

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.

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

Let \((A_i)_{i\in I}\) span \(M_{D}(\mathbb {C})\), and let \(C:\mathbb {C}^{m_2}\otimes \mathbb {C}^D\longrightarrow \mathbb {C}^{m_1}\otimes \mathbb {C}^D\) satisfy, for every \(i\in I\),

Then there is a matrix \(F:\mathbb {C}^{m_2}\to \mathbb {C}^{m_1}\) such that \(C=F\otimes \mathbb {1}_D\).

A matrix on \(\bigoplus _k \mathbb {C}^{D_k}\) is block diagonal if and only if every entry from one summand to a distinct summand is zero.

If a matrix on \(\bigoplus _k \mathbb {C}^{D_k}\) commutes with every block projection \(P_j\), then it is block diagonal.

If a matrix \(Z \in M_{D}(\mathbb {C})\) commutes with every element of \(M_{D}(\mathbb {C})\), then there exists \(\lambda \in \mathbb {C}\) such that \(Z = \lambda \mathbb {1}_D\).

For any words \(w_1, w_2\), \(A^{w_1 \cdot w_2} = A^{w_1} A^{w_2}\).

If \(w = (J_1, \ldots , J_m)\) is a word in the blocked alphabet, where each \(J_k\) is an \(L\)-tuple in \(\{ 0, \ldots , d{-}1\} ^L\), then concatenating the block words gives a word \(\widetilde{w}\) of length \(mL\) in the original alphabet and \((A^{[L]})^w = A^{\widetilde{w}}\).

For a word \(w\), the word evaluation of a block-diagonal tensor remains block diagonal:

If \(B^i = X A^i X^{-1}\) for all \(i\), then \(B^w = X A^w X^{-1}\) for every word \(w\).

For a configuration \(\sigma =(\sigma _0,\ldots ,\sigma _{N-1})\), let \(w_\sigma =[\sigma _0,\ldots ,\sigma _{N-1}]\). Then \(A^{w_\sigma } =A^{\sigma _0}A^{\sigma _1}\cdots A^{\sigma _{N-1}}\).

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for any word \(w\), \((\zeta A)^w = \zeta ^{|w|} A^w\), where \(|w|\) is the length of \(w\).

Let \(R\) be a ring, let \(V\) be an \(R\)-module, and let \(I\) and \(K\) be finite index sets. Let \(T\subseteq I\), and let \(\{ u_i\} _{i\in I}\) and \(\{ v_k\} _{k\in K}\) be vectors in \(V\). Suppose that, for every \(i\notin T\), there are a scalar \(\alpha _i\) and an index \(\phi (i)\in K\) such that \(u_i=\alpha _i v_{\phi (i)}\). If

and the coproduct-indexed family \(w\colon T\amalg K\to V\), defined by \(w(\operatorname {inl}(i))=u_i\) and \(w(\operatorname {inr}(k))=v_k\), is linearly independent, then \(a_i=0\) for every \(i\in T\).

If \(A\) and \(B\) are gauge equivalent and \(A\) is \(L\)-block injective, then \(B\) is \(L\)-block injective.

Gauge-phase equivalence is symmetric after identifying equal bond dimensions. Moreover, suppose that \(A\), \(B\), and \(C\) have identified bond dimensions, that \(X_1\) and \(X_2\) are invertible, that \(\zeta _1,\zeta _2\ne 0\), and that, for every physical index \(i\),

Then

so \(B\) and \(C\) are gauge-phase equivalent after the corresponding identification of their bond dimensions.

For each basis state \(\sigma = (\sigma _1, \ldots , \sigma _N)\) of the blocked system, where each \(\sigma _k\) is an \(L\)-tuple in \(\{ 0, \ldots , d{-}1\} ^L\), there exists a basis state \(\widetilde{\sigma } = (\widetilde{\sigma }_1, \ldots , \widetilde{\sigma }_{NL})\) of the original system such that \(V^{(N)}(A^{[L]})_\sigma = V^{(NL)}(A)_{\widetilde{\sigma }}\).

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for every \(N\) and every configuration \(\sigma = (i_1, \ldots , i_N)\), \(V^{(N)}(\zeta A)_\sigma = \zeta ^{N} V^{(N)}(A)_\sigma \).

Nonzero proportionality implies eventual nonzero proportionality, is symmetric, and equality of MPV families is the special case with scalar \(1\), at every positive length.

If the one-site matrices \(\{ A^i\} _{i=0}^{d-1}\) span \(M_{D}(\mathbb {C})\), then the length-one word products span \(M_{D}(\mathbb {C})\), so \(A\) is \(1\)-block injective.

\(O_{AB}(N) = \overline{\langle V^{(N)}(A) | V^{(N)}(B) \rangle }\).

If \(V^{(N)}(A)_\sigma = V^{(N)}(A')_\sigma \) and \(V^{(N)}(B)_\sigma = V^{(N)}(B')_\sigma \) for all \(N {\gt} 0\) and all \(\sigma \), then \(O_{AB}(N) = O_{A'B'}(N)\) for all positive \(N\).

Both periodic relations are equivalence relations (reflexive, symmetric, transitive).

Unit gauge-phase equivalence implies gauge-phase equivalence.

If \(A\) and \(B\) generate the same MPV family, then for any blocking length \(L\), the blocked tensors \(A^{[L]}\) and \(B^{[L]}\) also generate the same MPV family.

If \(A\) and \(B\) are gauge equivalent, then they generate the same MPV family.

If \(A\) and \(B\) are gauge equivalent and \(A\) is normal, then \(B\) is normal.

If \(B^i = \zeta X A^i X^{-1}\) for every physical index \(i\), then for every system size \(N\) and configuration \(\sigma \) one has \(V^{(N)}(B)_\sigma = \zeta ^N V^{(N)}(A)_\sigma \).

The MPV of a block-diagonal tensor decomposes as

Equivalently, for each configuration \(\sigma \) one has