Classical capacity of the amplitude damping channel

Derive a tractable, explicit expression for the classical capacity C(A_p) of the qubit amplitude damping channel A_p for arbitrary damping parameter p in [0,1].

Background

The classical capacity of a quantum channel is generally given by a regularized Holevo information, which is intractable to compute in most cases due to potential non-additivity. Although the Holevo information of the amplitude damping channel is known, the additivity properties required to simplify the capacity formula are not established, leaving the actual classical capacity unknown.

The paper develops approximation-based upper bounds on classical capacity for channels near classes with additive Holevo information, but the amplitude damping channel remains a prominent case where an exact capacity formula is lacking.

References

A well-known example of a qubit channel with unknown classical capacity is the amplitude damping channel. Deriving a tractable expression for its classical capacity remains a major open problem in quantum information theory.

— Approaches for approximate additivity of the Holevo information of quantum channels  (1709.01111 - Leditzky et al., 2017) in Section 1, Introduction

However, it is still unclear whether its known single-use rate is exact even with arbitrary entangled encodings.

— Classical Capacity and Entanglement Cost of the Amplitude Damping Channel  (2609.28592 - Tang et al., 23 Sep 2026) in Section 1, Introduction