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Smallest output dimension violating minimum output entropy additivity

Unsolved ID op_e724615844c52297 Last edited 25 September 2026
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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

Reports do not certify correctness or automatically change the problem's status. Progress policy.

  • 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).DOIarXiv
[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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BibTeX

@incollection{qiqcop_op_e724615844c52297,
  title = {Smallest output dimension violating minimum output entropy additivity},
  booktitle = {Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo)},
  year = {2026},
  howpublished = {\url{https://qiqc-op.com/problem/op_e724615844c52297/}},
  note = {Stable ID op_e724615844c52297; status: Unsolved; accessed 2026-10-08}
}

Plain text

“Smallest output dimension violating minimum output entropy additivity,” Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo), https://qiqc-op.com/problem/op_e724615844c52297/, ID op_e724615844c52297, accessed 2026-10-08.

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op_e724615844c52297
01M207QTTXDG3NHDWTRA9R0203