On the undecidability of quantum channel capacities

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.

arXiv comment29 pages, 2 figures, v2 contained an error in Section 4 which is corrected in v3 by updating Section 4 and Section 2. The introduction is also updated