On the undecidability of quantum channel capacities

TL;DR

This paper explores the computational complexity of quantum channel capacities, showing they are likely uncomputable or undecidable, with some capacities being QMA-hard or uncomputable.

Contribution

It formally demonstrates the hardness and potential undecidability of computing quantum channel capacities, advancing understanding of their computational limits.

Findings

Quantum capacity computation is QMA-hard.

Entanglement-assisted zero-error capacity is uncomputable.

Quantum channel capacities may be fundamentally undecidable.

Abstract

An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the entanglement-assisted zero-error capacity under some restrictions is uncomputable; indicative of the fact that quantum channel capacities may generally be undecidable.