Algebraic Geometry of Matrix Product States

TL;DR

This paper explores the algebraic structure of matrix product states (MPS) in quantum systems, providing polynomial equations that characterize MPS and linking them to classical trace algebra and hidden Markov models.

Contribution

It introduces polynomial equations characterizing translation-invariant MPS and connects MPS to trace algebras and hidden Markov models, offering new parameterizations and conjectures.

Findings

Explicit polynomial equations for small qubit systems

Connection between MPS and trace algebras/hidden Markov models

Four conjectures on MPS parameter identifiability

Abstract

We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states. For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions. Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS. We make four conjectures on the identifiability of MPS parameters.