# Purification and time-reversal deny entanglement in LOCC-distinguishable orthonormal bases

## TL;DR

This paper proves that in LOCC-distinguishable orthonormal bases, the states must be unentangled, using a simple argument based on purification and time-reversal, applicable in general probabilistic theories.

## Contribution

It introduces a straightforward proof that entangled states cannot be perfectly distinguished by LOCC within complete orthonormal bases, extending to arbitrary probabilistic theories.

## Key findings

- Entangled states cannot be perfectly distinguished by LOCC in such bases.
- The proof relies on properties of purity and time-reversal.
- Results apply broadly beyond quantum theory, to general probabilistic frameworks.

## Abstract

We give a simple proof, based on time-reversibility and purity, that a complete orthonormal family of pure states which can be perfectly distinguished by LOCC cannot contain any entangled state. Our results are really about the shape of certain states and processes, and are valid in arbitrary categorical probabilistic theories with time-reversal. From the point of view of the resource theory of entanglement, our results can be interpreted to say that free processes can distinguish between the states in a complete orthonormal family only when the states themselves are all free.