# Detecting mixed-unitary quantum channels is NP-hard

## TL;DR

Determining whether a quantum channel is mixed-unitary is computationally NP-hard, even approximately, highlighting the complexity of classifying quantum channels in quantum information theory.

## Contribution

This paper proves the NP-hardness of recognizing mixed-unitary quantum channels from their Choi representations, a significant complexity result in quantum information.

## Key findings

- NP-hardness of mixed-unitary channel detection
- Hardness persists even with approximate channels
- Implications for quantum channel classification

## Abstract

A quantum channel is said to be a mixed-unitary channel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channel, it is NP-hard with respect to polynomial-time Turing reductions to determine whether or not that channel is a mixed-unitary channel. This hardness result holds even under the assumption that the channel is not within an inverse-polynomial distance (in the dimension of the space upon which it acts) of the boundary of the mixed-unitary channels.