2 Matrix Product Vectors

Fix a physical dimension $d$ and a bond dimension $D$. This chapter studies the tensors that generate matrix product vector (MPV) families. All tensors here are translation-invariant with periodic boundary conditions (PBC).

2.1 Basic definitions

Definition 2.1.1 MPS tensor

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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,

$\begin{tenkz}[physical=up] \tn[up=$i$]{A} \end{tenkz}$

The black node denotes the tensor $A$, the horizontal legs are virtual, and the upper leg is the physical index $i$.

Definition 2.1.2 Word evaluation

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Given a word $w = (i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L$, the word evaluation is the matrix product

\begin{align} A^w & := A^{i_1} A^{i_2} \cdots A^{i_L} \in M_{D}(\mathbb {C}). \label{eq:mps_word_eval} \end{align}

The empty word evaluates to the identity, $A^\varnothing = \mathbb {1}_D$. In tensor-network notation,

$\begin{tenkz}[physical=up] \tn[up=$i_1$]{A}\tnspan[brace below]{4}{$L\text{ copies}$} & \tn{A} & \tndots & \tn[up=$i_L$]{A} \end{tenkz}$

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 $(i_1,\ldots ,i_L)$.

Lemma 2.1.3 Word evaluation along a listed configuration

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

Proof ▶

This follows by induction on $N$ from the recursive definition of word evaluation.

Lemma 2.1.4 Multiplicativity of word evaluation

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

Proof ▶

The claim follows from the definition of $A^w$.

Lemma 2.1.5 Centre of $M_{D}(\mathbb {C})$

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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 $Z = \lambda \mathbb {1}_D$.

Proof ▶

Since $Z$ commutes with every matrix unit, all off-diagonal entries of $Z$ vanish and all diagonal entries are equal.

Lemma 2.1.6 Rectangular amplified commutant

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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 $i\in I$,

\begin{align} (\mathbb {1}_{m_1}\otimes A_i)C & =C(\mathbb {1}_{m_2}\otimes A_i). \notag \end{align}

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

Proof ▶

Each $(a,b)$ multiplicity block of $C$ commutes with every $A_i$. Since the $A_i$ span the full matrix algebra, Lemma 2.1.5 makes this block a scalar multiple of the identity. These scalars are the entries of $F$.

Definition 2.1.7 Matrix product vector

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

\begin{align} |V^{(N)}(A)\rangle & = \sum _{i_1, \ldots , i_N = 0}^{d-1} \operatorname{tr}\! (A^{i_1} A^{i_2} \cdots A^{i_N}) |i_1, \ldots , i_N\rangle \in (\mathbb {C}^d)^{\otimes N}. \label{eq:mps_mpv_state} \end{align}

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

\begin{align} V^{(N)}(A)_\sigma & := \operatorname{tr}\! (A^{i_1} A^{i_2} \cdots A^{i_N}). \label{eq:mps_mpv_component} \end{align}

The coefficient function $\sigma \mapsto V^{(N)}(A)_\sigma $ gives the components of the vector in (??); 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

$\begin{tenkz}[periodic, physical=up] \tn[up=$i_1$]{A}\tnspan[brace below]{4}{$N\text{ copies}$} & \tn{A} & \tndots & \tn[up=$i_N$]{A} \end{tenkz}$

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

Definition 2.1.8 Transfer map

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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

\begin{align} \mathcal{E}_A(X) & = \sum _{i=0}^{d-1} A^i X (A^i)^\dagger . \label{eq:mps_transfer_map} \end{align}

Diagrammatically, the transfer map is the double-layer contraction

$\begin{tenkz}[sandwich, east={cup=$X$}] \tn{A} \\ \tn*{A} \end{tenkz}$

in which the upper node denotes $A$, the lower node denotes $A^\dagger $, and the physical index is summed over between them.

Remark 2.1.9

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The transfer map is automatically positive: if $X \ge 0$ then $\mathcal{E}_A(X) \ge 0$, because each summand $A^i X (A^i)^\dagger $ is positive semidefinite. The abstract positive-map framework appears in Chapter 4.

2.2 Site-periodic tensor families

Definition 2.2.1 Site-periodic tensor family

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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 $D$.

Definition 2.2.2 Periodic same-state and gauge-equivalence relations

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For site-periodic tensors $A,B$ at fixed period $m$:

equality of the induced site-periodic states;
cyclic gauge equivalence.

Lemma 2.2.3 Equivalence structures for periodic relations

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

2.3 Gauge equivalence and same MPV

Definition 2.3.1 Gauge equivalence

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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\} $,

\begin{align} B^i & = X A^i X^{-1}. \label{eq:mps_gauge_equiv} \end{align}

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

$\begin{tenkz}[physical=up] \tnX{X} & \tn[up=$i$]{A} & \tnX{X^{-1}} \end{tenkz}$

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.

Definition 2.3.2 Same MPV

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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 $N$.

Definition 2.3.3 Same MPV - different bond dimensions

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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.

Definition 2.3.4 Positive-length same MPV - different bond dimensions

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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 $N {\gt} 0$.

Remark 2.3.5

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Definition 2.3.2 is the same-bond-dimension specialization of Definition 2.3.3. We keep both because later results use both the same-dimension and different-dimension forms.

Definition 2.3.6 Proportional MPV

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Two tensors $A$ and $B$ (of possibly different bond dimensions) generate weakly proportional MPV families if for every $N{\gt}0$ there exists $c_N \in \mathbb {C}$ such that $|V^{(N)}(A)\rangle = c_N |V^{(N)}(B)\rangle $.

Definition 2.3.7 Nonzero proportional MPV

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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 ] .

Definition 2.3.8 Eventually nonzero proportional MPV

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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 F.1.10.

Lemma 2.3.9 Elementary nonzero proportionality rules

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

Proof ▶

Forgetting the nonvanishing condition gives weak proportionality. The eventual form follows by viewing an all-length statement as an eventual one. Symmetry follows by inverting the scalar $c_N$ at each length, and equality is the case $c_N = 1$.

Lemma 2.3.10 Scaling of word evaluation

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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 $w$.

Proof ▶

Induction on $w$: the base case gives $\zeta ^0 = 1$, and each step picks up one factor of $\zeta $.

Lemma 2.3.11 Scaling of MPV components

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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 $.

Proof ▶

Apply Lemma 2.3.10 to the word $w = (i_1, \ldots , i_N)$ of length $N$, then take the trace; the scalar $\zeta ^{N}$ pulls out by linearity.

Definition 2.3.12 Gauge-phase equivalence

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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 $i$,

\begin{align} B^i & = \zeta X A^i X^{-1}. \label{eq:mps_gauge_phase_equiv} \end{align}

Proof ▶

Gauge equivalence is the special case $\zeta = 1$.

Remark 2.3.13

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After spectral-radius normalization one may restrict $\zeta $ to a unit-modulus phase $e^{i\phi }$. We allow an arbitrary nonzero scalar because later canonical-form statements keep the block-scaling factors explicit.

Lemma 2.3.14 Gauge-phase equivalence across equal bond dimensions

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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 $i$,

\begin{align} B^i & =\zeta _1X_1A^iX_1^{-1},\notag \\ C^i & =\zeta _2X_2A^iX_2^{-1}. \notag \end{align}

Then

\begin{align} C^i & =(\zeta _2\zeta _1^{-1})(X_2X_1^{-1}) B^i(X_1X_2^{-1}), \label{eq:mps_gauge_phase_compose} \end{align}

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

Proof ▶

Symmetry follows by replacing $(X,\zeta )$ with $(X^{-1},\zeta ^{-1})$. Formula (??) follows by substituting the first gauge formula into the second.

Lemma 2.3.15 Gauge covariance of word evaluation

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

Proof ▶

Induction on the word length, using $X X^{-1} = \mathbb {1}$.

Theorem 2.3.16 Gauge invariance of MPVs

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

Proof ▶

Let $B^i = X A^i X^{-1}$. By Lemma 2.3.15, $B^w = X A^w X^{-1}$ for every word $w$. Then $\operatorname{tr}(B^w) = \operatorname{tr}(X A^w X^{-1}) = \operatorname{tr}(A^w)$ by cyclicity of the trace.

Theorem 2.3.17 Gauge-phase scaling of MPVs

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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 $.

Proof ▶

Combine gauge covariance of word evaluation with scalar scaling, then take traces.

2.4 Injectivity and normality

Definition 2.4.1 Injectivity

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

The inverse-tensor characterization of injectivity says that $A$ is injective exactly when there is a tensor $A^{-1}$ such that

\begin{align} \sum _i (A^i)_{\alpha ,\beta }(A^{-1,i})_{\alpha ',\beta '} & = \delta _{\alpha ,\alpha '}\delta _{\beta ,\beta '}. \label{eq:mps_injective_inverse} \end{align}

Thus (??) identifies the two pairs of virtual indices when $A^{-1}$ is contracted with $A$ through their common physical index. In shorthand, $A^{-1}A=\mathbb {1}$. This is the identity used in [ CPGSV16 , Section 2.3, lines 324–331 ] .

Definition 2.4.2 $L$-block injectivity

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A tensor $A$ is $L$-block injective if

\begin{align} \operatorname{span}_{\mathbb {C}}\{ A^{i_1} \cdots A^{i_L} : (i_1,\ldots ,i_L) \in \{ 0,\ldots ,d{-}1\} ^L\} & = M_{D}(\mathbb {C}). \label{eq:mps_block_injective} \end{align}

An injective tensor is $1$-block injective.

Lemma 2.4.3 One-site injectivity gives one-block injectivity

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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.

Proof ▶

The length-one words are exactly the letters $i \in \{ 0,\ldots ,d{-}1\} $, and evaluating a length-one word gives the matrix $A^i$. Thus the length-one word span is the same span as in one-site injectivity.

Definition 2.4.4 Normal tensor

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

Lemma 2.4.5 Gauge invariance of block injectivity

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If $A$ and $B$ are gauge equivalent and $A$ is $L$-block injective, then $B$ is $L$-block injective.

Proof ▶

Conjugation by an invertible matrix maps the span of the length-$L$ words of $A$ onto the span of the length-$L$ words of $B$. Since conjugation is an automorphism of the full matrix algebra, fullness of the first span implies fullness of the second.

Theorem 2.4.6 Gauge transport of normality

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If $A$ and $B$ are gauge equivalent and $A$ is normal, then $B$ is normal.

Proof ▶

Apply Lemma 2.4.5 at a blocking length for which $A$ is block-injective.

Remark 2.4.7

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The algebraic characterization of normality in Definition 2.4.4 — eventual full matrix span, equivalently eventual block injectivity — is equivalent to the spectral characterization used in [ CPGSV21 , Definition IV.1 ] : after TP normalization, the transfer map $\mathcal{E}_A$ is a primitive channel, equivalently irreducible with trivial peripheral spectrum. This equivalence is proved in [ SPGWC10 , Proposition 3 ] . In what follows, the directions of the equivalence are supplied by Lemma 8.7.2.1, Theorem 8.7.4.2 for primitive normalized tensors with positive-definite fixed point, the Burnside route through Theorem 8.8.2 for the aperiodic algebraic criterion, and Theorem 9.9.2.1. In particular, every subsequent use of “normal” here refers to this single notion; Definition 9.9.2 employs the spectral formulation, but it describes the same class of tensors.

2.5 Canonical form

Definition 2.5.1 Block-injective canonical form

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

a number of blocks $r \ge 1$,
bond dimensions $D_1, \ldots , D_r$,
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\} $,
scaling factors $\mu _1, \ldots , \mu _r \in \mathbb {C}$.

The associated tensor is the block-diagonal tensor

\begin{align} A^i & = \bigoplus _{k=1}^{r} \mu _k A_k^i. \label{eq:mps_canonical_form} \end{align}

This block-injective formulation of the canonical form is used in [ PGVWC07 ] ; see also [ CPGSV21 ] for the normal canonical form.

Remark 2.5.2

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Here we use the block-injective canonical form: each block is injective in the sense of Definition 2.4.1. This is stronger than the NT-block canonical form of [ CPGSV21 ] , where the blocks are only normal in the spectral sense. Chapter 9 treats the normal canonical form needed for the blocking-based multi-block theory, while the periodic irreducible-form extension is kept separate. This section isolates the stronger block-injective form used in the injective and block-injective arguments.

Definition 2.5.3 Block projection

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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.

Definition 2.5.4 Block-diagonal matrix

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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.

Lemma 2.5.5 Off-block characterization

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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.

Proof ▶

The forward direction follows from the definition of a direct sum of matrices. Conversely, if all off-block entries vanish, the diagonal blocks reconstruct the original matrix.

Lemma 2.5.6 Commutation with block projections

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

Proof ▶

Compare the $(i,j)$ block of the identity $XP_j=P_jX$ for $i\ne j$. The right multiplication by $P_j$ keeps that block, while the left multiplication by $P_j$ kills it. Hence every off-block entry is zero, and Lemma 2.5.5 applies.

Definition 2.5.7 Block-diagonal tensor from blocks

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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

\begin{align} A^i & = \bigoplus _{k=1}^{r} \mu _k A_k^i. \notag \end{align}

Lemma 2.5.8 Word evaluation of a block-diagonal tensor

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For a word $w$, the word evaluation of a block-diagonal tensor remains block diagonal:

\begin{align} A^w & = \bigoplus _k \mu _k^{|w|} A_k^w. \label{eq:mps_block_word_eval} \end{align}

Proof ▶

Induct on the word and multiply block-diagonal matrices block by block; every letter contributes one scalar factor $\mu _k$ on the $k$-th block.

Theorem 2.5.9 MPV decomposition

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The MPV of a block-diagonal tensor decomposes as

\begin{align} |V^{(N)}(A)\rangle & = \sum _{k=1}^{r} \mu _k^N |V^{(N)}(A_k)\rangle . \label{eq:mps_mpv_decomposition} \end{align}

Equivalently, for each configuration $\sigma $ one has

\begin{align} V^{(N)}(A)_\sigma & = \sum _{k=1}^{r} \mu _k^N V^{(N)}(A_k)_\sigma . \notag \end{align}

Proof ▶

By Lemma 2.5.8, the full word evaluation is block diagonal with blocks $\mu _k^N A_k^w$ for a word $w$ of length $N$. Taking the trace of a block-diagonal matrix gives the sum of the block traces: $\operatorname{tr}(A^w) = \sum _k \mu _k^N \operatorname{tr}(A_k^w)$.

2.6 Blocking

Definition 2.6.1 Blocked tensor

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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 $(i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L$:

\begin{align} (A^{[L]})^{(i_1, \ldots , i_L)} & = A^{i_1} A^{i_2} \cdots A^{i_L}. \label{eq:mps_blocked_tensor} \end{align}

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 $A^{[L]}$.

$\begin{tenkz}[physical=up, tensor style=box] \tn[up=$i_1$]{A}\tnspan[box, label pos=west]{4}{A^{[L]}} & \tn{A} & \tndots & \tn[up=$i_L$]{A} \end{tenkz}$

Lemma 2.6.2 Blocked word evaluation

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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 $(A^{[L]})^w = A^{\widetilde{w}}$.

Proof ▶

Induct on the blocked word $w$ and split off the first block word. Lemma 2.1.4 turns concatenation of blocks into multiplication of the corresponding evaluations.

Lemma 2.6.3 Blocking preserves MPV coefficients

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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 $V^{(N)}(A^{[L]})_\sigma = V^{(NL)}(A)_{\widetilde{\sigma }}$.

Proof ▶

Concatenate the block words and apply the blocked word evaluation lemma (Lemma 2.6.2).

Theorem 2.6.4 Blocking preserves same MPV

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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.

Proof ▶

By Lemma 2.6.3, each MPV coefficient of the blocked tensor $A^{[L]}$ equals a coefficient of $A$ at the corresponding concatenated configuration. The same holds for $B^{[L]}$ (with the same concatenated configuration). Since $A$ and $B$ agree on all configurations, so do $A^{[L]}$ and $B^{[L]}$.

2.7 MPV overlap

Definition 2.7.1 MPV as Hilbert-space vector

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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 $\mathbb {C}^{d^N}$.

Definition 2.7.2 MPV inner product

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The MPV inner product between two tensors $A$ and $B$ at system size $N$ is

\begin{align} \langle V^{(N)}(A) | V^{(N)}(B) \rangle & := \sum _{\sigma \in \{ 0,\ldots ,d{-}1\} ^N} \overline{V^{(N)}(A)_\sigma } V^{(N)}(B)_\sigma , \label{eq:mps_inner_product} \end{align}

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

Definition 2.7.3 MPV overlap

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The MPV overlap between two tensors $A$ (of bond dimension $D_1$) and $B$ (of bond dimension $D_2$) at system size $N$ is

\begin{align} O_{AB}(N) & := \sum _{\sigma \in \{ 0,\ldots ,d{-}1\} ^N} V^{(N)}(A)_\sigma \overline{V^{(N)}(B)_\sigma }. \label{eq:mps_overlap} \end{align}

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:

$\begin{tenkz}[rows={ket, bra}, periodic] \tn{A} & \tndots & \tn{A} \\ \tn*{B}\tnspan[brace below]{3}{$N\text{ sites}$} & \tndots & \tn*{B} \end{tenkz}$

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.

Lemma 2.7.4 Overlap equals conjugate inner product

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

Proof ▶

The overlap sums $V_\sigma \overline{W_\sigma }$ while the inner product sums $\overline{V_\sigma } W_\sigma $. Complex conjugation swaps these.

Lemma 2.7.5 Overlap determined by positive-length MPV

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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 $N$.

Proof ▶

Direct pointwise rewriting: if the MPV coefficients agree for all positive chain lengths, the overlap sums are identical.