Computability of the regularized capacity formulas

Ascertain whether the regularized multi-letter expressions defining the quantum capacity Q(Φ) and the private classical capacity P(Φ) of finite-dimensional quantum channels Φ are computable; specifically, determine whether there exists an algorithm that, given a description of Φ, outputs Q(Φ) or P(Φ) to arbitrary precision.

Background

Quantum and private classical capacities are defined via the Lloyd–Shor–Devetak (LSD) regularized formulas, which require taking limits of single-letter quantities over arbitrarily many channel uses. This regularization makes exact evaluation difficult and raises fundamental questions about their algorithmic computability.

The paper emphasizes that, despite extensive study, it is currently unknown whether these capacities are computable in general, citing prior work that establishes this uncertainty. Clarifying the computability status would have broad implications for quantum information theory and for practical capacity estimation.