A gap that survives combining channels
A mathematical construction describes finite-dimensional quantum channels generated by permutations whose one-copy minimum output entropy converges to a free-limit value. For two copies, the minimum output entropy eventually becomes strictly smaller than twice the one-copy value. Minimum output entropy is the least uncertainty a channel can produce at its output, so the result says that two uses can fall below the simple expectation of doubling the one-use figure.
This is a mathematical construction, not an empirical study. Its objects are permutation tuples indexed by Θ and acting on [N], rather than human, animal or other empirical specimens.
Separate jobs inside the construction
The paper applies strong convergence to random-channel free limits and uses permutation strong convergence, together with deterministic lifts, to build finite-dimensional channels. In this setting, strong convergence is the condition that keeps the finite permutation models aligned with the limiting spectral behavior needed for the entropy argument.
The design separates two tasks. One part approximates the one-copy spectrum or output body—the set of outputs available from the channel—while an auxiliary phase system fixes the output produced by an entangled input across two copies for every permutation tuple. This lets the one-copy approximation improve without altering the specially controlled two-copy output.
The target for the one-copy calculation is the free-compression channel. The paper identifies its output body and its minimum output entropy, giving the finite construction a precise benchmark for comparison.
The two-copy part has an exact feature. For every N at least 2 and every permutation tuple, a Bell-state input produces the same fixed mixture of the Bell state and the maximally mixed state. The identity holds for each finite tuple, rather than emerging only after a limiting argument.
From probability to an algorithm
The paper turns the approximation problem into an entropy test. If the one-copy output-body error is no greater than a chosen tolerance, the minimum output entropy is bounded below by the free-limit value minus a continuity penalty. If twice that penalty is smaller than the free-limit entropy defect, the strict two-copy inequality follows.
For independent uniform permutation tuples, the required admissibility condition is met with probability tending to 1 for every fixed positive tolerance. This is an asymptotic statement; the supplied result does not give a finite-size probability threshold.
When the free-limit entropy defect is positive, the random construction satisfies the strict minimum-output-entropy inequality with probability tending to 1. The result remains asymptotic and does not supply a practically usable finite size.
Under the paper’s positive-defect and tolerance condition, the O’Donnell–Wu lift algorithm returns a finite-dimensional nonadditivity counterexample in time polynomial in M for every sufficiently large M. The threshold for “sufficiently large” is not reported in the supplied statement.
A certified result at impractical scale
To obtain a finite existence estimate, the authors use the Chen–Garza-Vargas–Tropp–van Handel master inequalities to derive a one-sided upper-tail bound. The estimate is sufficient to establish existence, but the paper says the resulting certified instance is not practically executable.
The numerical entropy certificate leaves a positive but extremely small margin between twice the one-copy benchmark and the Bell-state value. That narrow separation is the numerical basis for the strict inequality, while also underscoring how much precision the construction requires.
The work therefore offers a systematic route for using permutations in place of more general random constructions and for asymptotically derandomizing the result. Its certified finite-dimensional example remains an existence guarantee at a scale the paper describes as impractical.
The source is an arXiv version 1 preprint dated 26 Aug 2026. Its disclosure says ChatGPT 5.6 Pro supplied the main proof ideas; the authors say they verified the statements and accepted responsibility, while AI also assisted with language editing and Python drafting. A permanent GitHub link is cited for the numerical calculations.
Paper data and sources
Original title: Superadditivity of classical communication over quantum channels via random and deterministic permutations
Authors: Benjamin Lovitz, Peixue Wu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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