Practical finite-dimensional counterexample construction

Construct a simple closed-form or practically computable finite-dimensional quantum-channel counterexample exhibiting nonadditivity of minimum output entropy using permutation-generated channels.

Background

The paper replaces Haar-random unitaries with random permutations and uses the deterministic O’Donnell–Wu procedure to establish an asymptotic derandomization. However, the resulting dimensions are enormous, and the construction does not yield a simple formula or an instance that can currently be computed or implemented in practice. This problem concerns turning the existential and asymptotic construction into a practically usable counterexample.

References

Thus the random construction can be derandomized in an asymptotic algorithmic sense, although finding a simple closed-form or practically computable counterexample remains open.

— Superadditivity of classical communication over quantum channels via random and deterministic permutations  (2608.25961 - Lovitz et al., 26 Aug 2026) in Abstract

Therefore, it remains a major challenge whether a practical search is successful.

— Superadditivity of classical communication over quantum channels via random and deterministic permutations  (2608.25961 - Lovitz et al., 26 Aug 2026) in Section 1, subsection “Sketch proof of the main result”

It remains an interesting problem to find simpler, smaller size quantum circuits. Due to limited time and resources, we leave those questions for future works and present this work as a proof of principle.

— Explicit channels with unbounded gains in classical communication using entangled inputs  (2609.26743 - Cheng et al., 22 Sep 2026) in Discussions section