Computational aspects of the trace norm contraction coefficient
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.
Cite
@article{arxiv.2507.16737,
title = {Computational aspects of the trace norm contraction coefficient},
author = {Idris Delsol and Omar Fawzi and Jan Kochanowski and Akshay Ramachandran},
journal= {arXiv preprint arXiv:2507.16737},
year = {2025}
}
Comments
23 pages, 2 figures; minor improvements and corrections