Computational aspects of the trace norm contraction coefficient

• URL: http://arxiv.org/abs/2507.16737v1
• Date: Tue, 22 Jul 2025 16:22:07 GMT
• Title: Computational aspects of the trace norm contraction coefficient
• Abstract summary: We show that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard.<n>We also establish a converging hierarchy of semidefinite programming upper bounds on the contraction coefficient.
• Score: 4.916646834691489
• License: http://creativecommons.org/licenses/by/4.0/
• 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 by solved efficiently. Our hardness results also hold for deciding if the contraction coefficient is equal to 1. As a consequence, we show that 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.

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