Smallest output dimension violating minimum output entropy additivity

Problem

What is the smallest output dimension in which the minimum output von Neumann entropy of quantum channels fails to be additive? For a density operator \(\sigma\) on a finite-dimensional Hilbert space write \(H(\sigma)=-\operatorname{Tr}(\sigma\log_2\sigma)\) for its von Neumann entropy, and for a quantum channel \(\Phi\) define its minimum output entropy by

\begin{equation} S_{\min}(\Phi):=\min_{\rho\in\mathcal D(A)}H\bigl(\Phi(\rho)\bigr), \tag{1} \end{equation}

where \(\mathcal D(A)\) is the set of density operators on the input space \(A\) of \(\Phi\). A product input is always available in \(\Phi\otimes\Psi\), so Eq. (1) gives \(S_{\min}(\Phi\otimes\Psi)\leq S_{\min}(\Phi)+S_{\min}(\Psi)\) for any two channels; the phenomenon at issue is the strict inequality

\begin{equation} S_{\min}(\Phi\otimes\Psi)<S_{\min}(\Phi)+S_{\min}(\Psi), \tag{2} \end{equation}

and a pair \((\Phi,\Psi)\) of channels satisfying Eq. (2) is called a violation. Define the output-dimension threshold

\begin{equation} d_{\min}:=\min\bigl\{d\in\mathbb N:\ \text{some violation }(\Phi,\Psi) \text{ has both output spaces of dimension at most }d\bigr\}, \tag{3} \end{equation}

with input and environment dimensions arbitrary. Determine the exact value of \(d_{\min}\) in Eq. (3). The companion constructive target, also open at every output dimension, is a practically computable, explicitly presented pair of finite-dimensional channels with a certified instance of Eq. (2): deterministic asymptotic algorithms now output violating pairs once their size parameter is sufficiently large, but none supplies a practically computable finite-dimensional instance.

Source

Hastings settled the qualitative existence of violations of Eq. (2) probabilistically [Has09]. The threshold formulation of Eq. (3) is derived from the quantitative dimension analysis of Belinschi, Collins, and Nechita [BCN16] together with the recorded constructive status at \(p=1\) — where deterministic asymptotic algorithms exist but no practically computable finite-dimensional realization is known [LLW26], [LovWu26], [ZZCW26]; no single source states it as a numbered open problem.

Progress

Hastings disproved additivity of the minimum output von Neumann entropy with finite-dimensional random channels, so the set in Eq. (3) is nonempty; the construction is probabilistic and yields no explicit pair [Has09], as are its later simplifications [BH10].
Belinschi, Collins, and Nechita computed the exact limiting minimum output entropy of a random channel in the conjugate-pair ensemble, exhibited violations of Eq. (2) with size approaching one bit, and proved that their Bell-state criterion detects a violation at output dimension \(183\) while failing almost surely up to dimension \(182\) within their ensemble; the bound \(d_{\min}\leq183\) in Eq. (3) follows, but the threshold is ensemble-specific [BCN16].
Leung, Lovitz, and Wu proved nonadditivity of the minimum output Rényi entropy for \(p\in[0,1/4)\cup(3/4,\infty]\) [LLW26]. At \(p=1\) all their arguments remain probabilistic; within their locally normalized projection ensemble, high-precision numerical optimization evaluates the Bell-state criterion as detecting a violation already at output dimension \(182\), against \(183\) for the ensemble criterion of [BCN16], a threshold they state is ensemble- and witness-specific and not universal over all channels. Deterministic asymptotic algorithms followed: Lovitz and Wu replace Haar randomness by permutations and derandomize via the O’Donnell–Wu construction, polynomial-time for fixed channel parameters and accuracy but with size estimates beyond practical reach [LovWu26], and Zhen, Zhu, Chen, and Wang, Theorem 2.1, deterministically output near-free permutation representations with a certified entropy gap for all sufficiently large target sizes [ZZCW26]; neither supplies a practically computed instance.
Counterexamples away from the von Neumann order \(p=1\) split by construction type: Hayden and Winter proved nonadditivity for every \(p>1\) using random channels [HW08], while the deterministic constructions are due to Grudka, Horodecki, and Pankowski for every \(p>2\) [GHP10] and to Derksen and Lovitz for every \(p>1\), from subspaces with high geometric measure of entanglement [DL26]. None of these produces a violation of Eq. (2) at the von Neumann order \(p=1\).
Krohn-Grimberghe certified, with exact rational witnesses, a strict violation of minimum output Rényi-entropy additivity for all \(0<p\leq1/22\) for the explicit four-to-three channel pair of Cubitt et al. [KG26], [CHLMW08]: explicit small-dimension violations exist at small Rényi orders, sharpening the contrast with the open \(p=1\) case.
Collins and Youn violated additivity of the regularized minimum output entropy through a non-random construction on an infinite-dimensional commuting-operator system and noted that it is not clear the approach works in finite dimensions, leaving Eq. (3) untouched [CY22].
King proved that unital qubit channels are additive with an arbitrary partner [Kin02], so a violation in Eq. (3) at \(d=2\) would require both channels to be non-unital; beyond the trivial \(d_{\min}\geq2\), reflecting that one-dimensional outputs are additive with every partner, no lower bound is known.

Comment

The unresolved gap is the range \(2\leq d_{\min}\leq183\) of Eq. (3): the rigorous upper bound is the ensemble-specific analysis of [BCN16], whose Bell-state criterion detects a violation at output dimension \(183\) and fails almost surely through \(182\) within that ensemble, while the threshold \(182\) of [LLW26] is the numerical evaluation, by high-precision optimization, of the corresponding criterion in a different, locally normalized ensemble — numerical evidence that a violation may already exist at output dimension \(182\), not a certified bound, so the certified upper endpoint remains \(183\); the lower bound is the trivial additivity of one-dimensional outputs, and the strongest structural constraint is that a \(d=2\) violation needs two non-unital channels [Kin02]. No practically computable finite-dimensional pair realizing Eq. (2) at \(p=1\) is known in any dimension: existence is probabilistic [Has09], [LLW26], and the deterministic algorithms of [LovWu26] and [ZZCW26] reach violations only asymptotically, without a practical instance; that constructive target is archived in the closed-form nonadditivity record. The sibling extremal question of the maximal violation size \(S_{\min}(\Phi)+S_{\min}(\Psi)-S_{\min}(\Phi\otimes\Psi)\) at fixed output dimension is not archived here. The delayed-onset additivity-violation problem poses the orthogonal onset question for a single channel and its tensor powers, so neither record subsumes the other.

References

[Has09]: M. B. Hastings, “Superadditivity of Communication Capacity Using Entangled Inputs,” Nature Physics 5, 255–257 (2009).DOI arXiv
[BCN16]: S. T. Belinschi, B. Collins, and I. Nechita, “Almost One Bit Violation for the Additivity of the Minimum Output Entropy,” Communications in Mathematical Physics 341(3), 885–909 (2016).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 \(0\leq p<1/4\),” arXiv preprint (2026).arXiv
[DL26]: H. Derksen and B. Lovitz, “Constructive Counterexamples to the Additivity of Minimum Output Rényi Entropy of Quantum Channels for All \(p>1\),” arXiv preprint (2026), version 2.arXiv
[GHP10]: A. Grudka, M. Horodecki, and L. Pankowski, “Constructive Counterexamples to Additivity of Minimum Output Rényi Entropy of Quantum Channels for All \(p>2\),” Journal of Physics A 43, 425304 (2010).arXiv
[CHLMW08]: T. Cubitt, A. W. Harrow, D. Leung, A. Montanaro, and A. Winter, “Counterexamples to Additivity of Minimum Output \(p\)-Rényi Entropy for \(p\) Close to 0,” Communications in Mathematical Physics 284(1), 281–290 (2008).arXiv
[KG26]: A. Krohn-Grimberghe, “Exact Certification of a Positive-Order Rényi Additivity Violation for an Explicit Channel Pair,” arXiv preprint (2026).arXiv
[CY22]: B. Collins and S.-G. Youn, “Additivity Violation of the Regularized Minimum Output Entropy,” Documenta Mathematica 27, 1299–1320 (2022).arXiv
[HW08]: P. Hayden and A. Winter, “Counterexamples to the Maximal \(p\)-Norm Multiplicativity Conjecture for All \(p>1\),” Communications in Mathematical Physics 284(1), 263–280 (2008).arXiv
[Kin02]: C. King, “Additivity for Unital Qubit Channels,” Journal of Mathematical Physics 43, 4641–4653 (2002).arXiv
[BH10]: F. G. S. L. Brandão and M. Horodecki, “On Hastings’ Counterexamples to the Minimum Output Entropy Additivity Conjecture,” Open Systems & Information Dynamics 17(1), 31–52 (2010).arXiv
[LovWu26]: B. Lovitz and P. Wu, “Superadditivity of Classical Communication over Quantum Channels via Random and Deterministic Permutations,” preprint (August 2026).arXiv
[ZZCW26]: G. Zhen, C. Zhu, R. Chen, and X. Wang, “Deterministic Minimum-Output-Entropy Nonadditivity via Haagerup’s Inequality and Near-Free Permutation Representations,” preprint (August 2026).arXiv

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