Concavity of entropy under thinning

Yu, Yaming; Johnson, Oliver

doi:10.1109/isit.2009.5205880

Cited by 15 publications

(27 citation statements)

References 13 publications

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“…The thinning has been introduced by Rényi [27] as a discrete analogue of the rescaling of a continuous real random variable. The thinning has been involved with this role in discrete versions of the central limit theorem [28]- [30] and of the Entropy Power Inequality [31], [32]. Most of these results require the ad hoc hypothesis of the ultra log-concavity (ULC) of the input state.…”

Section: Introductionmentioning

confidence: 99%

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

Palma

Trevisan

Giovannetti

2017

IEEE Trans. Inform. Theory

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Abstract-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 tradeoff region of the Gaussian quantum-limited attenuator and the classical capacity region of the Gaussian degraded quantum broadcast channel.Index Terms-Gaussian quantum channels, Gaussian quantum attenuator, thinning, von Neumann entropy, isoperimetric inequality.

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Section: Introductionmentioning

confidence: 99%

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

Palma

Trevisan

Giovannetti

2017

IEEE Trans. Inform. Theory

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“…Since then, progress has been limited. In [23], Yu and Johnson considered the thinning operation of Rényi [15], and proved a result which implies concavity of entropy when each p i (t) is proportional to t or 1 − t . Further, [10], Theorem 1.1, proved Theorem 1.2 in the case where each p i (t) is either constant or equal to t.…”

Section: Theorem 12 (Shepp-olkin Theorem) For Any N ≥ 1 the Functimentioning

confidence: 99%

A proof of the Shepp–Olkin entropy concavity conjecture

Hillion¹,

Johnson²

2017

Bernoulli

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This is the final published version of the article (version of record). It first appeared online via Project Euclid at http://projecteuclid.org/euclid.bj/1495505104. Please refer to any applicable terms of use of the publisher. University of Bristol -Explore Bristol Research General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms We prove the Shepp-Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof refines an argument previously presented by the same authors, which resolved the conjecture in the monotonic case (where all the parameters are simultaneously increasing). In fact, we show that the monotonic case is the worst case, using a careful analysis of concavity properties of the derivatives of the probability mass function. We propose a generalization of Shepp and Olkin's original conjecture, to consider Rényi and Tsallis entropies.

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“…Since then, progress was limited. In [106,Theorem 2] it was proved that the entropy is concave when for each i either p i (0) = 0 or p i (1) = 0. (In fact, this follows from the case n = 2 of Theorem 6.3 above).…”

Section: Entropy Concavity and The Shepp-olkin Conjecturementioning

confidence: 99%

Entropy and Thinning of Discrete Random Variables

Johnson

2017

Convexity and Concentration

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We describe five types of results concerning information and concentration of discrete random variables, and relationships between them, motivated by their counterparts in the continuous case. The results we consider are information theoretic approaches to Poisson approximation, the maximum entropy property of the Poisson distribution, discrete concentration (Poincaré and logarithmic Sobolev) inequalities, monotonicity of entropy and concavity of entropy in the Shepp-Olkin regime.

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Concavity of entropy under thinning

Cited by 15 publications

References 13 publications

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

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

A proof of the Shepp–Olkin entropy concavity conjecture

Entropy and Thinning of Discrete Random Variables

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