Computability of capacities of memoryless quantum channels

Determine whether the classical capacity and the quantum capacity of a fixed memoryless quantum channel—modeled as a completely positive trace-preserving map used independently across channel uses—are algorithmically uncomputable in the Turing sense.

Background

While capacities of communication channels with memory can be uncomputable, the computability status of capacities for memoryless quantum channels remains unresolved. The authors highlight this as a central open question.

Known coding theorems for quantum capacity involve regularizations over many channel uses, and no single-letter formula is available. Establishing (un)computability would clarify fundamental limits and the possibility of algorithmic evaluation of these capacities.