Additivity of minimum output entropy for qubit channels

The minimum output entropy of a quantum channel measures how pure its outputs can be kept. Using two channels side by side on an entangled input can push the joint output entropy below the sum of the two minima; Hastings proved this in 2009 for very large channels. Whether it can happen when one of the two channels acts on a single qubit, the smallest nontrivial case, is open.

Problem

A quantum channel $\mathcal N$ is a completely positive, trace-preserving linear map on matrices. Its minimum output entropy is

$$S_{\min}(\mathcal N)=\min_{\rho}S(\mathcal N(\rho)),\qquad S(\sigma)=-\operatorname{Tr}\sigma\log_2\sigma,$$

where $\rho$ ranges over input density matrices. Product inputs $\rho_1\otimes\rho_2$ show that $S_{\min}(\Phi\otimes\Omega)\le S_{\min}(\Phi)+S_{\min}(\Omega)$ for any two channels.

Question. Let $\Phi$ be any qubit channel, taking $2\times2$ density matrices to $2\times2$ density matrices, and let $\Omega$ be any channel between finite-dimensional systems. Is it always true that

$$S_{\min}(\Phi\otimes\Omega)=S_{\min}(\Phi)+S_{\min}(\Omega)\,?$$

The channel $\Phi$ need not be unital, that is $\Phi(I)\neq I$ is allowed; the unital case is solved [1]. A negative answer needs one pair $(\Phi,\Omega)$ and one input whose output entropy is strictly below $S_{\min}(\Phi)+S_{\min}(\Omega)$.

Background

Additivity of minimum output entropy was a central conjecture of quantum Shannon theory: Shor showed that, taken over all channels, it is equivalent to additivity of the Holevo capacity, which governs classical communication [3]. Hastings disproved it in 2009 with random channels in very high dimensions, which turned the question into one of finding the channels for which additivity does hold [3]. King settled every unital qubit channel, with an arbitrary partner, in 2002 [1]. In his 2015 survey Holevo noted that the non-unital qubit case still had no analytical proof, despite convincing numerical evidence [3].

What is known

• Additivity holds with an arbitrary partner when $\Phi$ is a unital qubit channel [1], and when $\Phi$ is entanglement-breaking, in any dimension [2].
• Qubit channels that map some pure state to a pure state, which include amplitude damping and every qubit channel with at most two Kraus operators: Tang, Zhu, Bai and Wang (2026, unrefereed preprint) prove a strong superadditivity inequality for the convex roof of the output entropy with any partner [4]. They use it for Holevo-capacity additivity; applied to pure inputs, it gives additivity of minimum output entropy.
• Generalized amplitude-damping channels, a two-parameter family that models a qubit exchanging energy with a thermal environment; with positive damping at finite temperature they have no pure output [5]. Fang (2026, unrefereed preprint) proves that their maximal output $p$-norm is multiplicative with any completely positive partner for every $p\ge1$ [5], and multiplicativity for $p$ near $1$ implies additivity of minimum output entropy [3]. A September 2026 preprint from OpenAI reports a proof of minimum-output-entropy additivity for these channels with any finite-dimensional partner [6]. It has not been independently verified, and its Lean formalization covers the capacity and the additivity of Holevo information over repeated uses of the same channel, not additivity with a different partner [6].
• Still open: non-unital qubit channels that are not entanglement-breaking, have no pure output and are not generalized amplitude damping up to unitary rotations of input and output.

What would count as progress

• A proof for a larger non-unital family, for example all qubit channels that commute with rotations about one axis (phase-covariant channels), which contain generalized amplitude damping [5].
• The weaker self-product statement $S_{\min}(\Phi\otimes\Phi)=2S_{\min}(\Phi)$ for every non-unital qubit channel $\Phi$.
• A counterexample: explicit Kraus operators for $\Phi$ and $\Omega$, an input state, and rigorous lower bounds on both minima. The qubit minimum is an optimization over the Bloch sphere; bounding $S_{\min}(\Omega)$ for a small partner can be certified with interval arithmetic.

References

1. C. King, "Additivity for unital qubit channels", Journal of Mathematical Physics 43, 4641–4653 (2002). doi:10.1063/1.1500791
2. P. W. Shor, "Additivity of the classical capacity of entanglement-breaking quantum channels", Journal of Mathematical Physics 43, 4334–4340 (2002). doi:10.1063/1.1498000
3. A. S. Holevo, "Gaussian optimizers and the additivity problem in quantum information theory", Russian Mathematical Surveys 70, 331–367 (2015). doi:10.1070/RM2015v070n02ABEH004949
4. Z. Tang, C. Zhu, G. Bai and X. Wang, "Classical capacity and entanglement cost of the amplitude damping channel", preprint (2026). arXiv:2609.28592
5. K. Fang, "The classical capacity of generalized amplitude-damping channels and its strong-converse exponent", preprint (2026). arXiv:2609.14621
6. OpenAI, "Classical capacity and entropy inequalities for generalized amplitude damping", OpenAI math release preprint (2026). Paper, Lean scope

Source

Imported from QIQCOP Zoo op_eba8dda5ff69d0f2, curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.