Computational aspects of the trace norm contraction coefficient

TL;DR

This paper proves that approximating the trace norm contraction coefficient of quantum channels is NP-hard, highlighting fundamental computational challenges in quantum information theory unlike classical cases.

Contribution

It establishes NP-hardness results for key quantum information problems and introduces a hierarchy of semidefinite programming bounds for the contraction coefficient.

Findings

Approximating the contraction coefficient is NP-hard.

Deciding if the contraction coefficient equals 1 is NP-hard.

A converging hierarchy of SDP upper bounds is developed.

Abstract

We show that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard. Equivalently, this shows that determining the optimal success probability for encoding a bit in a quantum system undergoing noise is NP-hard. This contrasts with the classical analogue of this problem that can clearly be solved efficiently. We also establish the NP-hardness of deciding if the contraction coefficient is equal to 1, i.e., the channel can perfectly preserve a bit. As a consequence, deciding if a non-commutative graph has an independence number of at least 2 is NP-hard. In addition, we establish a converging hierarchy of semidefinite programming upper bounds on the contraction coefficient.