Minimum output entropy additivity for qubit channels

Problem

Is minimum output von Neumann entropy additive whenever one channel is a qubit channel? Let \(\Phi:\mathcal L(\mathbb C^2)\to\mathcal L(\mathbb C^2)\) be any completely positive trace-preserving map, and let \(\Omega:\mathcal L(A)\to\mathcal L(B)\) be any finite-dimensional quantum channel. For a channel \(\mathcal N\), define

\begin{equation} S_{\min}(\mathcal N):=\min_{\rho\in\mathcal D(A_{\mathcal N})}S(\mathcal N(\rho)),\qquad S(\sigma):=-\operatorname{Tr}\sigma\log_2\sigma, \tag{1} \end{equation}

where \(\mathcal D(A_{\mathcal N})\) is the set of density operators on the input space of \(\mathcal N\). Prove or disprove

\begin{equation} S_{\min}(\Phi\otimes\Omega)=S_{\min}(\Phi)+S_{\min}(\Omega)\qquad\text{for every such pair }(\Phi,\Omega). \tag{2} \end{equation}

In Eq. (2), both the input and output of \(\Phi\) have dimension two; no unitality assumption \(\Phi(I_2)=I_2\) is imposed. Since product inputs are admissible in the minimization of Eq. (1), a counterexample must certify strict subadditivity for a specific finite-dimensional pair.

Source

Holevo explicitly records the unresolved nonunital-qubit case in Section 2.5, p. 13 of the arXiv version, in a discussion where the second channel is arbitrary [Hol15]. Hayashi reviews the qubit numerical evidence and the unital theorem in Section 9.9.1, pp. 552–553 [Hay17]. Ruskai notes the unresolved nonunital self-power possibility immediately after Problem 19, p. 16 [Rus07].

Progress

King’s Theorem 1 proves the equality in Eq. (2) for every unital qubit channel with an arbitrary channel as partner. It also proves maximal output Schatten-\(p\) norm multiplicativity for every \(p\geq1\) in this unital setting [Kin02].
King and Koldan’s Theorem 1 proves maximal output Schatten-\(p\) norm multiplicativity for arbitrary qubit channels at \(p=2\) and \(p\geq4\), with an arbitrary finite-dimensional completely positive partner. These orders do not establish the von Neumann-entropy equality in Eq. (2) [KK06].
Large-dimensional nonadditivity results do not settle the qubit restriction. Leung, Lovitz, and Wu’s July 2026 preprint obtains random-channel counterexamples for Rényi orders \(p>3/4\) and \(0\leq p<1/4\). At \(p=1\), its Section 5.1 numerically evaluates an ensemble-specific Bell-input threshold at output dimension \(182\); this is neither a qubit counterexample nor a proof of a universal dimension threshold [LLW26].
Shor proves minimum-output-entropy additivity for an entanglement-breaking channel tensored with an arbitrary channel. This covers entanglement-breaking qubit channels, including nonunital examples, but leaves general nonunital qubit channels open [Sho02].

Comment

Audited on 2026-09-09 against the full source statements and later nonadditivity literature. The nonunital qubit case remains unresolved. The smallest-output-dimension record permits arbitrary input dimensions and bounds both output dimensions, whereas this question fixes one channel to have qubit input and output and leaves its partner unrestricted. The closed-form-witness record permits arbitrary dimensions, so a solution there need not answer this question. Two related unresolved variants are Holevo-capacity additivity with an arbitrary partner and minimum-output-entropy additivity on self-powers of a qubit channel. A general partner counterexample need not yield a qubit self-power counterexample. The universal additivity equivalences in unrestricted dimensions do not justify identifying the Holevo and minimum-output-entropy questions with these dimension constraints.

References

[Hay17]: M. Hayashi, Quantum Information Theory: Mathematical Foundation, 2nd ed., Graduate Texts in Physics, Springer (2017).DOI
[Hol15]: A. S. Holevo, “Gaussian optimizers and the additivity problem in quantum information theory,” Russian Mathematical Surveys 70(2), 331–367 (2015).DOI arXiv
[Rus07]: M. B. Ruskai, “Open Problems in Quantum Information Theory,” preprint (2007).arXiv
[Kin02]: C. King, “Additivity for unital qubit channels,” Journal of Mathematical Physics 43, 4641–4653 (2002).DOI arXiv
[KK06]: C. King and N. Koldan, “New multiplicativity results for qubit maps,” Journal of Mathematical Physics 47(4), 042106 (2006).DOI arXiv
[LLW26]: D. Leung, B. Lovitz, and P. Wu, “Counterexamples to additivity of minimum output \(p\)-Rényi entropy of quantum channels for \(p>3/4\) and \(0\leq p<1/4\),” preprint (2026).arXiv
[Sho02]: P. W. Shor, “Additivity of the Classical Capacity of Entanglement-Breaking Quantum Channels,” Journal of Mathematical Physics 43, 4334–4340 (2002).DOI arXiv

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