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arXiv:2206.02133

Proof of the Gaussian maximizers conjecture for the communication capacity of noisy heterodyne measurements

A. S. Holevo, S. N. Filippov

Published Jun 5, 2022• 2 citations
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Abstract

Basing on recently developed convex programming framework in the paper [arXiv:2204.10626], we provide a proof for a long-standing conjecture on optimality of Gaussian encondings for the ultimate communication rate of generalized heterodyne receivers under the oscillator energy constraint. Our results generalize previous ones (obtained under the assumption of validity of the energy threshold condition) and show a drastic difference in the structure of the optimal encoding within and beyond this condition. The core of the proof in the case beyond the threshold is a new log-Sobolev type inequality, which relates the generalized Wehrl entropy with the wavefunction gradient.

DOI
10.48550/arxiv.2206.02133

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Basing on recently developed convex programming framework in the paper [arXiv:2204.10626], we provide a proof for a long-standing conjecture on optimality of Gaussian encondings for the ultimate communication rate of generalized heterodyne receivers under the oscillator energy constraint. Our results generalize previous ones (obtained under the assumption of validity of the energy threshold condition) and show a drastic difference in the structure of the optimal encoding within and beyond this condition. The core of the proof in the case beyond the threshold is a new log-Sobolev type inequality, which relates the generalized Wehrl entropy with the wavefunction gradient.
In the present paper we give proof that the information-transmission capacity of the approximate position measurement with the oscillator energy constraint, which underlies noisy Gaussian homodyning in quantum optics, is attained on Gaussian encoding. The proof is based on general principles of convex programming. Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of the celebrated log-Sobolev inequality. We hope that this method should work also for other models lying out of the scope of the "threshold condition" ensuring that the upper bound for the capacity as a difference between the maximum and the minimum output entropies is attainable.
A long-standing problem in quantum Shannon theory is the classical capacity of bosonic Gaussian channels of various kinds [6].The Hypothesis of Gaussian Maximizers (HGM) states that the full capacity of such channels is attained at Gaussian encodings.A breakthrough was made in the papers [1] and [2], where HGM was proved for the important class of multimode gauge co-or contra-variant Gaussian channels [4].In [8] and [5] the solution was extended to a much broader class of channels satisfying a certain 'threshold condition'.At the same time the HGM remains open for a large variety of Gaussian channels lying beyond the scope of the threshold condition [7].In this note we sketch a novel approach to such problems, based on principles of convex programming, and illustrate it using the characteristic case of an approximate position measurement with energy constraint, underlying noisy Gaussian homodyning in quantum optics.Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of the celebrated log-Sobolev inequality.Let S be the convex set of all density operators in a separable Hilbert space H of the quantum system and X and Y be standard measurable spaces.Consider a measurement channel M : ρ → p ρ (y) = Tr ρm(y), where ρ ∈ S and m(y) is a uniformly bounded positive operator-valued function of y ∈ Y such that m(y) µ(dy) = I (I is the identity operator on H and µ is a measure).The encoding E = {π(dx), ρ(x)} is a probability measure π(dx) with a measurable family of
Abstract In this work we apply the general convex programming approach of [A. S. Holevo, Lobachevskii J. Math. 43(7), 1646–1650 (2022)] to the problem of minimization of convex closure of the output differential entropy of quantum measurement channel, involved in computation of the classical capacity of the channel, and representing noise in the channel. We show that in the bosonic Gaussian case the optimality conditions derived from the convex programming approach lead to formulation of some new generalizations of the famous logarithmic Sobolev inequality which we call ‘‘homodyne’’ and ‘‘heterodyne’’ inequalities. We also show that the ‘‘heterodyne’’ inequality implies the ‘‘homodyne’’ one.
A long-standing problem on the classical capacity of bosonic Gaussian channels has recently been resolved by proving the minimum output entropy conjecture. It is also known that the ultimate capacity quantified by the Holevo bound can be achieved asymptotically by using an infinite number of channels. However, it is less understood to what extent the communication capacity can be reached if one uses a finite number of channels, which is a topic of practical importance. In this paper, we study the capacity of Gaussian communication, i.e., employing Gaussian states and Gaussian measurements to encode and decode information under a single-channel use. We prove that the optimal capacity of single-channel Gaussian communication is achieved by one of two well-known protocols, i.e., coherent-state communication or squeezed-state communication, depending on the energy (number of photons) as well as the characteristics of the channel. Our result suggests that the coherent-state scheme known to achieve the ultimate information-theoretic capacity is not a practically optimal scheme for the case of using a finite number of channels. We find that overall the squeezed-state communication is optimal in a small-photon-number regime whereas the coherent-state communication performs better in a large-photon-number regime.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
This paper establishes a general theory of energy-constrained quantum and private capacities of quantum channels. We begin by defining various energy-constrained communication tasks, including quantum communication with a uniform energy constraint, entanglement transmission with an average energy constraint, private communication with a uniform energy constraint, and secret key transmission with an average energy constraint. We develop several code conversions, which allow us to conclude non-trivial relations between the capacities corresponding to the above tasks. We then show how the regularized, energy-constrained coherent information is equal to the capacity for the first two tasks and is an achievable rate for the latter two tasks, whenever the energy observable satisfies the Gibbs condition of having a well-defined thermal state for all temperatures and the channel satisfies a finite output-entropy condition. For degradable channels satisfying these conditions, we find that the single-letter energy-constrained coherent information is equal to all of the capacities. We finally apply our results to degradable quantum Gaussian channels and recover several results already established in the literature (in some cases, we prove new results in this domain). Contrary to what may appear from some statements made in the literature recently, proofs of these results do not require the solution of any kind of minimum output entropy conjecture or entropy photon-number inequality.