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arXiv:2608.25961

Superadditivity of classical communication over quantum channels via random and deterministic permutations

Benjamin Lovitz, Peixue Wu

quant-phmath-phmath.PRSubmitted Aug 26, 2026• Announced Aug 27, 2026• 0 citations• Revised Oct 5, 2026 (v2)
v1 · Aug 26, 2026v2 · Oct 5, 2026
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Abstract

Since Hastings' proof of superadditivity of classical communication over quantum channels, considerable effort has been devoted to finding a structural explanation of this phenomenon that was originally established by concentration of measure for Haar random unitaries. The main observation of this work is that Haar randomness can be replaced by random permutations without changing the limiting geometry responsible for nonadditivity. This replacement turns a continuous problem over unitary matrices into a discrete combinatorial problem over zero--one permutation matrices, and thereby opens a path toward derandomization. The theorem of Bordenave and Collins shows that random permutations have the required limiting behavior and the algorithm of O'Donnell and Wu then provides a deterministic asymptotic construction, running in polynomial time in the size when the channel parameters and accuracy are fixed. Thus the random construction can be derandomized in an asymptotic algorithmic sense. Finally, a quantitative random permutation estimate by Chen, Garza-Vargas, Tropp and van Handel gives a fully numerical estimate: there exists a tuple of 57,836,025 permutations acting on a set of size \[ N \le 5.422\times 10^{116216}\] such that the associated finite dimensional channel exhibits nonadditivity. This enormous value remains an obstacle to a practical construction.

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Since Hastings' proof of superadditivity of classical communication over quantum channels, considerable effort has been devoted to finding a structural explanation of this phenomenon that was originally established by concentration of measure for Haar random unitaries.
The main observation of this work is that Haar randomness can be replaced by random permutations without changing the limiting geometry responsible for nonadditivity. This replacement turns a continuous problem over unitary matrices into a discrete combinatorial problem over zero--one permutation matrices, and thereby opens a path toward derandomization. The theorem of Bordenave and Collins shows that random permutations have the required limiting behavior and the algorithm of O'Donnell and Wu then provides a deterministic asymptotic construction, running in polynomial time in the size when the channel parameters and accuracy are fixed. Thus the random construction can be derandomized in an asymptotic algorithmic sense.
Finally, a quantitative random permutation estimate by Chen, Garza-Vargas, Tropp and van Handel gives a fully numerical estimate: there exists a tuple of 57,836,025 permutations acting on a set of size [ N \le 5.422\times 10^{116216}] such that the associated finite dimensional channel exhibits nonadditivity. This enormous value remains an obstacle to a practical construction.
In a breakthrough, Hastings showed that there exist quantum channels whose classical Holevo capacity is superadditive i.e. more classical information can be transmitted by quantum encoding strategies entangled across multiple channel uses as compared to unentangled quantum encoding strategies. Hastings’ proof used Haar random unitaries to exhibit superadditivity. In this paper we show that a unitary chosen uniformly at random from an approximaten2/3n^{2/3}n2/3-design gives rise to a quantum channel with superadditive classical Holevo capacity, wherennnis the dimension of the unitary exhibiting the Stinespring dilation of the channel superoperator. We follow the geometric functional analytic approach of Aubrun, Szarek and Werner in order to prove our result. More precisely we prove a sharp Dvoretzky-like theorem stating that, with high probability under the choice of a unitary from an approximatettt-design, random subspaces of large dimension make a Lipschitz function take almost constant value. Such theorems were known earlier only for Haar random unitaries. We obtain our result by appealing to Low’s technique for proving concentration of measure for an approximatettt-design, combined with a stratified analysis of the variational behaviour of Lipschitz functions on the unit sphere in high dimension. The stratified analysis is the main technical advance of this work. Haar random unitaries require at leastΩ(n2)\Omega (n^{2})Ω(n2)random bits in order to describe them with good precision. In contrast, there exist exactn2/3n^{2/3}n2/3-designs using onlyO(n2/3log⁡n)O(n^{2/3} \log n)O(n2/3logn)random bits. Thus, our work can be viewed as a partial derandomisation of Hastings’ result, and a step towards the quest of finding an explicit quantum channel with superadditive classical Holevo capacity. Finally we also show that for anyp>1p > 1p>1, approximate unitaryn1.7n^{1.7}n1.7-designs give rise to channels violating subadditivity of Rényippp-entropy. In addition to stratified analysis, the proof of this result uses a new technique of approximating a monotonic differentiable function defined on a closed bounded interval and its derivative by moderate degree polynomials which should be of independent interest.
Correlations across channel uses can improve quantum communication rates, making optimization over arbitrarily large blocks a central difficulty in determining quantum capacity. We show how full permutation invariance limits this advantage for every finite-dimensional memoryless channel. The optimized coherent information per use converges to the single-use maximum and consequently attains its supremum at a finite block length. Thus finite-block superadditivity remains possible, while the asymptotic optimization reduces to a single channel use. We also establish a strong converse for pure-source entanglement generation with permutation-invariant inputs: at any fixed rate above the single-use coherent-information maximum, the fidelity tends to zero. Beyond these communication bounds, we identify a common structure underlying completely bounded norms and the sandwiched Rényi channel quantities governing discrimination against a replacer and entanglement-assisted communication. De Finetti reduction and permutation covariance yield systematic alternative proofs of their known multiplicativity and additivity results within a shared variational framework. We further show that, for permutation-invariant inputs, correlations cannot sustain an asymptotic reduction in output entropy per use below the single-use minimum. This entropy limit holds for Rényi orders greater than one and for the von Neumann entropy, extending the role of symmetry from communication bounds to the control of output entropy.
In this work we give an efficient construction of unitary kkk-designs using O~(k⋅poly(n))\tilde{O}(k\cdot poly(n))O~(k⋅poly(n)) quantum gates, as well as an efficient construction of a parallel-secure pseudorandom unitary (PRU). Both results are obtained by giving an efficient quantum algorithm that lifts random permutations over S(N)S(N)S(N) to random unitaries over U(N)U(N)U(N) for N=2nN=2^nN=2n. In particular, we show that products of exponentiated sums of S(N)S(N)S(N) permutations with random phases approximately match the first 2Ω(n)2^{Ω(n)}2Ω(n) moments of the Haar measure. By substituting either O~(k)\tilde{O}(k)O~(k)-wise independent permutations, or quantum-secure pseudorandom permutations (PRPs) in place of the random permutations, we obtain the above results. The heart of our proof is a conceptual connection between the large dimension (large-NNN) expansion in random matrix theory and the polynomial method, which allows us to prove query lower bounds at finite-NNN by interpolating from the much simpler large-NNN limit. The key technical step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-NNN expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension NNN.
We demonstrate superadditivity of one-shot zero-error classical capacity in an asymmetric communication setting where a noisy classical channel is used in parallel with a perfect quantum channel. Each channel individually supports only a fixed number of perfectly distinguishable messages. Their joint use enables transmission of strictly more messages than permitted by the product of the individual capacities. We present explicit constructions achieving this enhancement and establish that replacing the perfect quantum channel with a perfect classical channel eliminates the effect. Finally, we identify a structural criterion on the noisy channel governing this effect and show that the quantum advantage is rooted in Kochen-Specker contextuality.
We give an explicit non-random example of nonadditivity of minimum output von Neumann entropy. The proof uses a finite-dimensional construction which imitates free Haar unitary behavior. The same channel gives a violation of additivity for the minimum output Rényi-ppp entropy for any p∈[1,∞]p\in[1,\infty]p∈[1,∞]. Additionally, given any p0>0p_0>0p0​>0 and g>0\mathfrak g>0g>0, we extend the construction to obtain an explicit non-random channel which has entropy gap ≥g\ge\mathfrak g≥g uniformly over p∈[p0,∞]p\in[p_0,\infty]p∈[p0​,∞].
In this paper, we investigate the additivity phenomenon in the quantum dynamic capacity region of a quantum channel for trading the resources of classical communication, quantum communication, and entanglement. Understanding such an additivity property is important if we want to optimally use a quantum channel for general communication purposes. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity region, and it is largely unknown if this could lead to a strictly superadditive double or triple resource capacity region, respectively. For example, if a channel has additive classical and quantum capacities, can the classical-quantum capacity region be strictly superadditive? In this paper, we answer such questions affirmatively. We give proof-of-principle requirements for these channels to exist. In most cases, we can provide an explicit construction of these quantum channels. The existence of these superadditive phenomena is surprising in contrast to the result that the additivity of both classical-entanglement and classical-quantum capacity regions imply the additivity of the triple resource capacity region for a given channel.