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 [ CPGSV16 ; CPGSV21 ] , with the single-block injective case anchored in the earlier representation theorem of [ PGVWC07 ] .

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 equations ?? and ?? 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 equation ?? and the weight comparison then combine into a single gauge conjugacy of the two weighted block-diagonal tensors in equations ?? and ??. For a single injective block this reduces to the familiar conclusion \(B^i = X A^i X^{-1}\) for all physical indices \(i\), the single-block specialization of equation ??.

Chapters 2–11 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, 5, 6, and 7 develop the positive-map, multiplicative-domain, Perron–Frobenius, and overlap-decay tools used to control normal blocks. Chapters 8 and 9 supply the blocking, primitivity, irreducibility, and canonical-form reductions. Chapter 9 includes the invariant-projection splitting. Chapter 10 supplies bases of normal tensors and the power-sum comparison.

Chapter 12 uses the Fundamental Theorem to obtain virtual symmetry gauges, projective bond representations, and string-order criteria; its Kraus-mixing step appeals only to the later channel-representation chapter, not to any additional input for the Fundamental Theorem.

Beyond Chapter 12, the later chapters develop parent-Hamiltonian consequences, correlation estimates, examples, channel representations, normal forms, quantum dynamical semigroups, operator convexity, and entropy. They also include the standard examples of positive maps that fail to be completely positive. The final three chapters introduce matrix product operators and density operators, develop the renormalization fixed points of pure matrix product states, and then treat the zero-correlation-length and fixed-point conditions for mixed states.