The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have Gaussian Maximizers

We determine the p->q norms of the Gaussian one-mode quantum-limited attenuator and amplifier and prove that they are achieved by Gaussian states, extending to noncommutative probability the seminal theorem "Gaussian kernels have only Gaussian maximizers" (Lieb in Invent Math 102(1):179-208, 1990). The quantum-limited attenuator and amplifier are the building blocks of quantum Gaussian channels, which play a key role in quantum communication theory since they model in the quantum regime the attenuation and the noise affecting any electromagnetic signal. Our result is crucial to prove the longstanding conjecture stating that Gaussian input states minimize the output entropy of one-mode phase-covariant quantum Gaussian channels for fixed input entropy. Our proof technique is based on a new noncommutative logarithmic Sobolev inequality, and it can be used to determine the p->q norms of any quantum semigroup.Comment: Annales Henri Poincar\'e (2018

Passive states optimize the output of bosonic Gaussian quantum channels

An ordering between the quantum states emerging from a single mode gauge-covariant bosonic Gaussian channel is proven. Specifically, we show that within the set of input density matrices with the same given spectrum, the element passive with respect to the Fock basis (i.e. diagonal with decreasing eigenvalues) produces an output which majorizes all the other outputs emerging from the same set. When applied to pure input states, our finding includes as a special case the result of A. Mari, et al., Nat. Comm. 5, 3826 (2014) which implies that the output associated to the vacuum majorizes the others

Gaussian States Minimize the Output Entropy of the One-Mode Quantum Attenuator

We prove that Gaussian thermal input states minimize the output von Neumann entropy of the one-mode Gaussian quantum-limited attenuator for fixed input entropy. The Gaussian quantum-limited attenuator models the attenuation of an electromagnetic signal in the quantum regime. The Shannon entropy of an attenuated real-valued classical signal is a simple function of the entropy of the original signal. A striking consequence of energy quantization is that the output von Neumann entropy of the quantum-limited attenuator is no more a function of the input entropy alone. The proof starts from the majorization result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is based on a new isoperimetric inequality. Our result implies that geometric input probability distributions minimize the output Shannon entropy of the thinning for fixed input entropy. Moreover, our result opens the way to the multimode generalization, that permits to determine both the triple trade-off region of the Gaussian quantum-limited attenuator and the classical capacity region of the Gaussian degraded quantum broadcast channel

On the Entropy of Sums of Bernoulli Random Variables via the Chen-Stein Method

This paper considers the entropy of the sum of (possibly dependent and non-identically distributed) Bernoulli random variables. Upper bounds on the error that follows from an approximation of this entropy by the entropy of a Poisson random variable with the same mean are derived. The derivation of these bounds combines elements of information theory with the Chen-Stein method for Poisson approximation. The resulting bounds are easy to compute, and their applicability is exemplified. This conference paper presents in part the first half of the paper entitled "An information-theoretic perspective of the Poisson approximation via the Chen-Stein method" (see:arxiv:1206.6811). A generalization of the bounds that considers the accuracy of the Poisson approximation for the entropy of a sum of non-negative, integer-valued and bounded random variables is introduced in the full paper. It also derives lower bounds on the total variation distance, relative entropy and other measures that are not considered in this conference paper.Comment: A conference paper of 5 pages that appears in the Proceedings of the 2012 IEEE International Workshop on Information Theory (ITW 2012), pp. 542--546, Lausanne, Switzerland, September 201

Log-concavity and the maximum entropy property of the Poisson distribution

We prove that the Poisson distribution maximises entropy in the class of ultra-log-concave distributions, extending a result of Harremo\"{e}s. The proof uses ideas concerning log-concavity, and a semigroup action involving adding Poisson variables and thinning. We go on to show that the entropy is a concave function along this semigroup.Comment: 16 pages: revised version, accepted by Stochastic Processes and their Application

Surfactant films in lyotropic lamellar (and related) phases: Fluctuations and interactions

The analogy between soap films thinning under border capillary suction and lamellar stacks of surfactant bilayers dehydrated by osmotic stress is explored, in particular in the highly dehydrated limit where the soap film becomes a Newton black film. The nature of short-range repulsive interactions between surfactant-covered interfaces and acting across water channels in both cases will be discussed.Comment: 29 pages, 11 figures. Accepted for publication (2017/06/21), Advances in Colloid and Interface Scienc

Consistency of Bayes estimators of a binary regression function

When do nonparametric Bayesian procedures ``overfit''? To shed light on this question, we consider a binary regression problem in detail and establish frequentist consistency for a certain class of Bayes procedures based on hierarchical priors, called uniform mixture priors. These are defined as follows: let $\nu$ be any probability distribution on the nonnegative integers. To sample a function $f$ from the prior $\pi^{\nu}$, first sample $m$ from $\nu$ and then sample $f$ uniformly from the set of step functions from $[0,1]$ into $[0,1]$ that have exactly $m$ jumps (i.e., sample all $m$ jump locations and $m+1$ function values independently and uniformly). The main result states that if a data-stream is generated according to any fixed, measurable binary-regression function $f_0\not\equiv1/2$, then frequentist consistency obtains: that is, for any $\nu$ with infinite support, the posterior of $\pi^{\nu}$ concentrates on any $L^1$ neighborhood of $f_0$. Solution of an associated large-deviations problem is central to the consistency proof.Comment: Published at http://dx.doi.org/10.1214/009053606000000236 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org