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IEEE Transactions on Information Theory

Approximate Unitary Designs Give Rise to Quantum Channels With Super Additive Classical Holevo Capacity

Aditya Nema, Pranab Sen

Published Apr 21, 2022• 1 citation
Publisher

Abstract

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 approximate$n^{2/3}$-design gives rise to a quantum channel with superadditive classical Holevo capacity, where$n$is 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 approximate$t$-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 approximate$t$-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 least$\Omega (n^{2})$random bits in order to describe them with good precision. In contrast, there exist exact$n^{2/3}$-designs using only$O(n^{2/3} \log n)$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 any$p > 1$, approximate unitary$n^{1.7}$-designs give rise to channels violating subadditivity of Rényi$p$-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.

DOI
10.1109/tit.2022.3167604

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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.
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≤5.422×10116216 N \le 5.422\times 10^{116216}N≤5.422×10116216 such that the associated finite dimensional channel exhibits nonadditivity. This enormous value remains an obstacle to a practical construction.
The famous Johnson-Lindenstrauss lemma states that for any set of n vectors, there is a linear transformation into a space of dimension O(log n) that approximately preserves all their lengths. In fact, a Haar random unitary transformation followed by projection onto the first O(log n) coordinates followed by a scaling works as a valid transformation with high probability. In this work, we show that the Haar random unitary can be replaced by a uniformly random unitary chosen from a finite set called an approximate unitary t-design for t = O(log n). Choosing a unitary from such a design requires only polylogarithmic random bits as opposed to exponential in dimension random bits required to choose a Haar random unitary with reasonable precision. Moreover, since such unitaries can be efficiently implemented in the superpositional setting, our result can be viewed as an efficient quantum Johnson-Lindenstrauss transform akin to efficient quantum Fourier transforms widely used in earlier work on quantum algorithms. We prove our result by leveraging a method of Low for showing concentration for approximate unitary t-designs. We discuss algorithmic advantages and limitations of our result and conclude with a toy application to private information retrieval.
We prove a new concentration result for non-catalytic decoupling by showing that, for suitably largettt, applying a unitary chosen uniformly at random from an approximatettt-design on a quantum system followed by a fixed quantum operation almost decouples, with high probability, the given system from another reference system to which it may initially have been correlated. Earlier works either did not obtain high decoupling probability, or used provably inefficient unitaries, or required catalytic entanglement for decoupling. In contrast, our approximate unitary designs always guarantee decoupling with exponentially high probability and, under certain conditions, lead to computationally efficient unitaries. As a result we conclude that, under suitable conditions, efficiently implementable approximate unitary designs achieve relative thermalisation in quantum thermodynamics with exponentially high probability. We also show the scrambling property of black hole, when the black hole evolution is according to pseudorandom approximate unitaryttt-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.
We prove a new concentration result for non-catalytic decoupling by showing that, for suitably large ttt, applying a unitary chosen uniformly at random from an approximate ttt-design on a quantum system followed by a fixed quantum operation almost decouples, with high probability, the given system from another reference system to which it may initially have been correlated. Earlier works either did not obtain high decoupling probability, or used provably inefficient unitaries, or required catalytic entanglement for decoupling. In contrast, our approximate unitary designs always guarantee decoupling with exponentially high probability and, under certain conditions, lead to computationally efficient unitaries. As a result we conclude that, under suitable conditions, efficiently implementable approximate unitary designs achieve relative thermalisation in quantum thermodynamics with exponentially high probability. We also show the scrambling property of black hole, when the black hole evolution is according to pseudorandom approximate unitary ttt-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.
The entanglement-assisted capacity of a quantum channel admits an additive single-letter characterization, implying that joint encodings across channel uses cannot increase the ultimate communication rate. Here, we show that this additive picture does not extend to communication reliability. Specifically, we prove that the Petz-Rényi channel information can be strictly superadditive for every α∈[1/2,1)α\in[1/2,1)α∈[1/2,1), yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.
We study collective state preparation and channel encoding in fermionic systems under parity superselection and composition governed by the canonical anticommutation relations. For a two-mode Gaussian state τττ, we obtain the exact entanglement of formation, EFphys(τ⊗m)=⌈m/2⌉log⁡2E_{\mathrm F}^{\mathrm{phys}}(τ^{\otimes m})=\lceil m/2\rceil\log2EFphys​(τ⊗m)=⌈m/2⌉log2 for every m≥1m\geq1m≥1. Two copies have the same formation value as one, violating additivity and strong superadditivity, while regularization halves the single-copy value. For an associated Gaussian channel, we determine the minimum output entropy and Holevo information at every blocklength and prove nonadditivity of both. Together, these results give exact answers to the fermionic counterparts of the four additivity questions associated with Shor and Hastings. One physical use carries no classical information, whereas two uses transmit one bit perfectly. The unassisted classical and zero-error classical capacities both equal one half bit per mode. A sharp finite-block bound gives a strong converse: the decoding success probability vanishes at every rate above capacity. These optimizations cover all physical pure-state decompositions and channel inputs, with Gaussian ensembles and classical codes attaining the optima. The formation and minimum output entropy formulas hold for every Rényi order α∈(0,∞]α\in(0,\infty]α∈(0,∞]. We also determine the unassisted quantum and zero-error quantum capacities, both equal to one half logical qubit per mode, and prove a strong converse: entanglement fidelity vanishes at every rate above capacity. For a noisy Gaussian family, we determine the one- and two-copy formation values and prove strict nonadditivity at full rank. For τττ, Bell violation under local parity-preserving measurements first appears at three copies.