Additivity of regularized minimum output entropy

Determine whether the regularized minimum output entropy is additive for arbitrary quantum channels, equivalently whether the classical capacity satisfies C(Φ ⊗ Ψ) = C(Φ) + C(Ψ) for all quantum channels Φ and Ψ.

Background

The paper establishes one-shot minimum-output-entropy violations for finite-dimensional channels but does not resolve the corresponding regularized question. Through the standard relationship between regularized minimum output entropy and classical communication capacity, the unresolved issue is expressed as additivity of the classical capacity under tensor products of quantum channels.

References

Evaluating the regularized formulas for concrete nonadditive channels remains an important problem.

— No information transmission through quantum channels above capacity  (2609.08998 - Cheng et al., 8 Sep 2026) in Section 2, subsection “Continuity of the Rényi capacities and the exponential strong converse” (immediately after Theorem 2)

However, it remains unclear whether the regularized minimum output entropy is additive, equivalently, whether we have \begin{equation} C(\Phi \otimes \Psi) = C(\Phi) + C(\Psi) \end{equation} where $C$ is the classical capacity of quantum channels~eq:intro-classical-capacity.

eq:intro-classical-capacity:

C(Φ)=lim⁡m→∞1mχ(Φ⊗m)=sup⁡m≥11mχ(Φ⊗m).C(\Phi) = \lim_{m\to\infty}\frac1m\chi(\Phi^{\otimes m}) = \sup_{m\geq1}\frac1m\chi(\Phi^{\otimes m}).

— Superadditivity of classical communication over quantum channels via random and deterministic permutations  (2608.25961 - Lovitz et al., 26 Aug 2026) in Section 1, subsection “Discussions and organization,” first bullet