A version 2 arXiv preprint dated 5 Oct 2026 reports that, for sufficiently large constructions, finite-dimensional quantum channels can have a two-copy minimum output entropy lower than twice their one-copy value. Minimum-output entropy is the smallest von Neumann entropy among a channel’s possible outputs, so the result says the least-entropic output from two copies can fall below the benchmark obtained by adding the two separate one-copy minima. The same theorem states that the finite-channel minimum output entropy converges to the value of an infinite-dimensional free-limit channel. The onset size for the strict inequality is not specified.
The underlying question is whether that infinite-dimensional model can be approximated by finite-dimensional channels without losing this nonadditive behavior. The mathematical objects are a family of channels indexed by tuples of permutations. One family consists of independent uniform random permutation tuples, while another is produced deterministically. The comparison is therefore between finite mathematical constructions and the infinite-dimensional model, rather than between experimental groups.
From an infinite model to finite channels
The construction assigns separate jobs to two families of quantum transformations. Clock unitaries enforce the two-copy estimate exactly, while permutation unitaries approximate the one-copy estimate. For every permutation tuple, the finite-dimensional channel exactly reproduces the diagonal-state output of the free-compression channel. That exact two-copy feature leaves the approximation argument focused on the one-copy side.
From random to deterministic
Randomness supplies the first route to finite examples. For independent uniform permutation tuples, the one-sided approximation error is below any fixed tolerance with probability tending to one as the permutation-set size grows. The statement is asymptotic, and it does not supply a success probability for a particular finite size.
The paper then gives a deterministic replacement for the random construction. For all sufficiently large input sizes, a deterministic polynomial-time algorithm produces a finite-dimensional channel satisfying the strict two-copy nonadditivity inequality. The numerical threshold at which that guarantee begins is not stated, so the result gives an eventual construction rather than a concrete size specification.
The numbers make the scale clear
The numerical bound makes the gap between existence and practicality stark. The abstract reports a tuple of 57,836,025 permutations and a set-size bound written in the available version as 5.422 × 10116216. The exponent notation is flattened in that text, and the figure is presented as a sufficient bound rather than a claim that the size is minimal.
For one selected parameterization, a high-precision calculation reports a value called E of 4.788343820071748458..., a second quantity called H_Bell of 9.576687612235380095..., and a positive margin of approximately 2.7908116821 × 10−8 entropy units. The margin is enough to support the strict nonadditivity inequality in that numerical construction, but it is tied to that parameter choice rather than presented as a universal gap.
The associated finite channel has output dimension 195. Its environment dimension is 58,429,215 (NNA − 1), while its input dimension is 1,460,730,375 (NNA − 1). Those figures, combined with the enormous sufficient bound, explain why the construction is mathematically finite but not practically executable.
The result stops short of a capacity theorem
The result also stops short of a statement about regularized classical capacity. The paper leaves open whether regularized minimum-output entropy is additive, equivalently whether the classical capacity of a product of two quantum channels equals the sum of their individual capacities. In short, the work establishes finite-dimensional minimum-output-entropy nonadditivity, not a resolution of the corresponding capacity question.
Taken together, the analysis gives an asymptotic route from random permutation tuples to deterministic finite-dimensional channels. Its immediate payoff is an existence theorem at a very large stated scale, not a practically executable instance. The size bound is only sufficient, and the capacity question remains open.
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
Original paper · Full text