Detecting mixed-unitary quantum channels is NP-hard

Understand

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.