1 Introduction

The central result is the Fundamental Theorem for translation-invariant matrix product vectors. The basic object is a family of matrices \(A=\{ A^i\} _{i=0}^{d-1}\), which assigns to each chain length \(N\) the vector \(|V^{(N)}(A)\rangle \) with coefficients indexed by \(\sigma =(i_1,\ldots ,i_N)\),

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

The main argument, for translation-invariant tensors, follows the canonical-form and basis-of-normal-tensors argument of [ 13 , 17 ] , with the single-block injective case anchored in the earlier representation theorem of [ 228 ] .

Notation

\(\mathbb {C}\), \(\mathbb {N}\), \(\mathbb {Z}\)	complex numbers, natural numbers, integers
\(M_{D}(\mathbb {C})\), \(\mathrm{GL}_D(\mathbb {C})\)	complex matrices and invertible complex matrices
\(\operatorname{tr}(X)\), \(\operatorname{span}_{\mathbb {C}}(S)\)	trace and complex linear span
\(A=\{ A^i\} _{i=0}^{d-1}\)	a translation-invariant MPS tensor
\(w=(i_1,\ldots ,i_L)\), \(A^w\)	a word and the product \(A^{i_1}\cdots A^{i_L}\)
\(\|V^{(N)}(A)\rangle \)	the length-\(N\) matrix product vector
\(V^{(N)}(A)_\sigma \)	the coefficient of \(\|V^{(N)}(A)\rangle \) at \(\sigma \)
\(O_{AB}(N)\)	the bilinear overlap \(\sum _\sigma V^{(N)}(A)_\sigma \, \overline{V^{(N)}(B)_\sigma }\)
\(\mathcal{E}_A(X)\)	the transfer map \(\sum _i A^i X(A^i)^\dagger \)
\(A^{[L]}\)	the length-\(L\) blocked tensor

In canonical form one writes

\begin{align} A^i & = \bigoplus _{j=1}^{g} \mu _j A_j^i, \label{eq:intro_canonical_form_a} \\ B^i & = \bigoplus _{k=1}^{h} \nu _k B_k^i. \label{eq:intro_canonical_form_b} \end{align}

The blocks in (2) and (3) are normal tensors. If \(|V^{(N)}(A)\rangle =|V^{(N)}(B)\rangle \) for every positive chain length \(N\), then the bases of normal tensors have the same size. After relabelling the blocks of \(B\), each matched pair satisfies

\begin{align} B_j^i & = \zeta _j X_j A_j^i X_j^{-1}, \label{eq:intro_matched_pair_gauge} \end{align}

for some invertible matrix \(X_j\) and nonzero scalar \(\zeta _j\), and the equal case compares the coefficient power sums attached to the relabelled bases. The block gauges in (4) and the weight comparison then combine into a single gauge conjugacy of the two weighted block-diagonal tensors in (2) and (3). For a single injective block this reduces to the conclusion \(B^i = X A^i X^{-1}\) for all physical indices \(i\), the single-block specialization of (4).

Chapters 2–9 carry out this argument. Chapter 2 fixes the matrix product vector notation, word evaluations, transfer maps, gauge equivalence, injectivity, blocking, and site-periodic tensor families. Chapter 3 proves the single-block theorem by trace pairings, linear extension, simplicity of matrix algebras, and Skolem–Noether. Chapters 4 and 5 develop the Perron–Frobenius and overlap-decay tools used to control normal blocks. The general theory of Schwarz inequalities and multiplicative domains is developed in the companion quantum-channel volume [ 216 , Chapter “Schwarz Inequalities and Multiplicative Domains” ] . Chapters 6 and 7 supply the blocking, primitivity, irreducibility, and canonical-form reductions. Chapter 7 includes the invariant-projection splitting. Chapter 8 supplies bases of normal tensors and the power-sum comparison.

Chapter 10 uses the Fundamental Theorem to obtain virtual symmetry gauges, projective bond representations, and string-order criteria; its Kraus-mixing step uses the rectangular Kraus-freedom theorem from the companion quantum-channel volume [ 216 , “Rectangular Kraus freedom” ] .

Beyond Chapter 10, subsequent chapters develop parent Hamiltonians (Chapter 11), correlation decay (Chapter 14), concrete examples (Chapter 15), matrix product density operators (Chapter 16), renormalization fixed points (Chapters 17 and 20), matrix product unitaries (Chapter 18), periodic tensors (Chapter 22), the algebraic Fundamental Theorem (Chapter 23), projected entangled pair states, their examples, \(G\)-injective tensors, and regional ground spaces (Chapters 24, 25, 26, and 27), and the asymmetric Fundamental Theorem (Chapter 29). General quantum-channel representations, positive-map examples, dynamical semigroups, operator convexity, and entropy are developed in the companion quantum-channel volume [ 216 , Chapters “Quantum Channels and Positive Maps,” “Channel Representations and Normal Forms,” “Positive but Not Completely Positive Maps,” “Operator Convexity and Jensen Inequalities,” “Quantum Dynamical Semigroups,” and “Quantum Entropy” ] .