Computability of quantum and private classical capacities

Ascertain whether the quantum capacity Q(Φ) and the private classical capacity P(Φ) of a general quantum channel Φ, defined by multi-letter regularized expressions, are computable functions.

Background

The paper recalls that the quantum capacity and the private classical capacity are given by regularized, multi-letter formulas. While these characterizations are foundational, evaluating them is notoriously difficult. The authors emphasize a fundamental algorithmic uncertainty regarding the computability of these quantities, citing recent work that highlights this issue. This open problem underpins the motivation for developing efficiently computable upper bounds, such as those provided by approximate degradability.