Smallest output dimension for a minimum-output-entropy violation

Feeding an entangled input into two quantum channels used side by side can make the joint output entropy smaller than the sum of the channels' minimum output entropies. Every known example needs large output dimensions. The smallest dimension in which this can happen is only known to lie between 2 and 182.

Problem

For a quantum channel $\Phi$ (a completely positive, trace-preserving map) with finite-dimensional input, let

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

over input density matrices $\rho$. Product inputs give $S_{\min}(\Phi\otimes\Psi)\le S_{\min}(\Phi)+S_{\min}(\Psi)$ for every pair of channels. Call the pair a violation if the inequality is strict. Let $d_{\min}$ be the least $d$ for which some violation has both output spaces of dimension at most $d$; input and environment dimensions are unrestricted.

Question. What is $d_{\min}$?

An upper bound $d_{\min}\le d$ needs one violation with outputs of dimension at most $d$, either explicit or proved to exist. A lower bound needs additivity for every pair whose outputs are smaller.

Background

Hastings disproved additivity of minimum output entropy in 2009 using random channels [1]. Such proofs give existence, not examples, and Fukuda, King and Moser's estimates of the dimensions needed reached about $10^{32}$ [2]. Belinschi, Collins and Nechita then computed the exact asymptotics of the standard test, a random channel paired with its complex conjugate on a maximally entangled input, and found violations from output dimension 183 [3]. Holevo's 2015 survey notes that it remains unclear what happens in small dimensions [2]. The threshold question is formulated by the source from these results. A counterexample to additivity for qubit channels with a partner whose output is also a qubit would give $d_{\min}=2$.

What is known

• $d_{\min}\le183$: the test with a random channel, its complex conjugate and the maximally entangled input detects violations at output dimension 183 and, almost surely, fails below it; for large outputs the violation approaches one bit [3]. The threshold came from a numerical study of an explicit function [3].
• $d_{\min}\le182$: Leung, Lovitz, Nechita and Wu (2026, unrefereed preprint revised in October) use random projection-induced channels and certify, with outward-rounded interval arithmetic, that random channel pairs with output dimension 182 violate additivity [4].
• All of these are probabilistic. Shou and Gorshkov (2026, unrefereed preprint) give the first explicit non-random example: a channel $\Phi$ with output dimension $2^{32}$ and input dimension $3\cdot2^{5\,268\,390\,106\,224\,805\,281\,791\,998}$ satisfying $2S_{\min}(\Phi)-S_{\min}(\Phi\otimes\Phi)>3/2^{34}$ [5].
• The source records no lower bound beyond the trivial $d_{\min}\ge2$: with one-dimensional outputs every entropy is zero.

What would count as progress

• A violation with both output dimensions at most 181. An explicit one comes as Kraus or Stinespring matrices, an input state, and lower bounds on $S_{\min}(\Phi)$ and $S_{\min}(\Psi)$ certified by interval arithmetic, checkable by a short program; an existence proof in the style of [3, 4] would also do.
• An explicit violation small enough to verify by computer, far below the dimensions in [5].
• A proof that every pair with both outputs of dimension 2 is additive, which would give $d_{\min}\ge3$.

References

1. M. B. Hastings, "Superadditivity of communication capacity using entangled inputs", Nature Physics 5, 255–257 (2009). doi:10.1038/nphys1224
2. A. S. Holevo, "Gaussian optimizers and the additivity problem in quantum information theory", Russian Mathematical Surveys 70, 331–367 (2015). doi:10.1070/RM2015v070n02ABEH004949
3. 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, 885–909 (2016). doi:10.1007/s00220-015-2561-z
4. D. Leung, B. Lovitz, I. Nechita and P. Wu, "Counterexamples to additivity of minimum output $p$-Rényi entropy of quantum channels for all $p\ge0$", preprint (2026). arXiv:2607.15210
5. L. Shou and A. V. Gorshkov, "A constructive violation of additivity of minimum output von Neumann entropy", preprint (2026). arXiv:2609.23946

Source

Imported from QIQCOP Zoo op_e724615844c52297, 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.