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.

35 pages, 2 figures; minor improvements and corrections. Computational model added. Accepted for publication in IEEE Transactions on Information Theory, DOI: https://doi.org/10.1109/TIT.2026.3743741

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2507.16737 [quant-ph]

(or arXiv:2507.16737v3 [quant-ph] for this version)

https://doi.org/10.1109/TIT.2026.3743741 DOI(s) linking to related resources