# Matrix Product Representation of Locality Preserving Unitaries

## TL;DR

This paper characterizes 1D locality-preserving unitaries using matrix product operators, providing a complete representation, a method to extract the GNVW index, and exploring cases where unitarity is size-dependent and non-locality-preserving.

## Contribution

It establishes that matrix product operators that are unitary exactly represent 1D locality-preserving unitaries and introduces fixed point conditions for their characterization.

## Key findings

- Matrix product unitaries preserve locality and represent all locality-preserving unitaries.
- A method to extract the GNVW index from matrix product representations.
- Examples of size-dependent unitaries that are non-local and have fractional indices.

## Abstract

The matrix product representation provides a useful formalism to study not only entangled states, but also entangled operators in one dimension. In this paper, we focus on unitary transformations and show that matrix product operators that are unitary provides a necessary and sufficient representation of 1D unitaries that preserve locality. That is, we show that matrix product operators that are unitary are guaranteed to preserve locality by mapping local operators to local operators while at the same time all locality preserving unitaries can be represented in a matrix product way. Moreover, we show that the matrix product representation gives a straight-forward way to extract the GNVW index defined in Ref.\cite{Gross2012} for classifying 1D locality preserving unitaries. The key to our discussion is a set of `fixed point' conditions which characterize the form of the matrix product unitary operators after blocking sites. Finally, we show that if the unitary condition is relaxed and only required for certain system sizes, the matrix product operator formalism allows more possibilities than locality preserving unitaries. In particular, we give an example of a simple matrix product operator which is unitary only for odd system sizes, does not preserve locality and carries a `fractional' index as compared to their locality preserving counterparts.