# Matrix Product States: Entanglement, symmetries, and state transformations

## TL;DR

This paper investigates the entanglement properties, symmetries, and state transformations of translationally-invariant matrix product states (MPS), providing criteria for state convertibility and classifying their symmetry structures.

## Contribution

It introduces a criterion for SLOCC transformations between MPS, classifies SLOCC equivalence classes for simple MPS, and characterizes all global and local symmetries of MPS.

## Key findings

- Established a criterion for SLOCC convertibility of MPS.
- Classified SLOCC classes for basic MPS.
- Characterized all symmetries of MPS, including inhomogeneous ones.

## Abstract

We analyze entanglement in the family of translationally-invariant matrix product states (MPS). We give a criterion to determine when two states can be transformed into each other by SLOCC transformations, a central question in entanglement theory. We use that criterion to determine SLOCC classes, and explicitly carry out this classification for the simplest, non-trivial MPS. We also characterize all symmetries of MPS, both global and local (inhomogeneous). We illustrate our results with examples of states that are relevant in different physical contexts.