14 research outputs found
Concavity of entropy under thinning
Building on the recent work of Johnson (2007) and Yu (2008), we prove that
entropy is a concave function with respect to the thinning operation T_a. That
is, if X and Y are independent random variables on Z_+ with ultra-log-concave
probability mass functions, then H(T_a X+T_{1-a} Y)>= a H(X)+(1-a)H(Y), 0 <= a
<= 1, where H denotes the discrete entropy. This is a discrete analogue of the
inequality (h denotes the differential entropy) h(sqrt(a) X + sqrt{1-a} Y)>= a
h(X)+(1-a) h(Y), 0 <= a <= 1, which holds for continuous X and Y with finite
variances and is equivalent to Shannon's entropy power inequality. As a
consequence we establish a special case of a conjecture of Shepp and Olkin
(1981).Comment: To be presented at ISIT0
A proof of the Shepp-Olkin entropy monotonicity conjecture
Consider tossing a collection of coins, each fair or biased towards heads,
and take the distribution of the total number of heads that result. It is
natural to conjecture that this distribution should be 'more random' when each
coin is fairer. Indeed, Shepp and Olkin conjectured that the Shannon entropy of
this distribution is monotonically increasing in this case. We resolve this
conjecture, by proving that this intuition is correct. Our proof uses a
construction which was previously developed by the authors to prove a related
conjecture of Shepp and Olkin concerning concavity of entropy. We discuss
whether this result can be generalized to -R\'{e}nyi and -Tsallis
entropies, for a range of values of .Comment: 16 page
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
-interpolation of probability measures on graphs
We generalize an equation introduced by Benamou and Brenier, characterizing
Wasserstein W_p-geodesics for p > 1, from the continuous setting of probability
distributions on a Riemannian manifold to the discrete setting of probability
distributions on a general graph. Given an initial and a final distributions
f_0 and f_1, we prove the existence of a curve (f_t) satisfying this
Benamou-Brenier equation. We also show that such a curve can be described as a
mixture of binomial distributions with respect to a coupling that is solution
of a certain optimization problem.Comment: 25 page
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
Monotonicity, thinning and discrete versions of the Entropy Power Inequality
We consider the entropy of sums of independent discrete random variables, in
analogy with Shannon's Entropy Power Inequality, where equality holds for
normals. In our case, infinite divisibility suggests that equality should hold
for Poisson variables. We show that some natural analogues of the Entropy Power
Inequality do not in fact hold, but propose an alternative formulation which
does always hold. The key to many proofs of Shannon's Entropy Power Inequality
is the behaviour of entropy on scaling of continuous random variables. We
believe that R\'{e}nyi's operation of thinning discrete random variables plays
a similar role to scaling, and give a sharp bound on how the entropy of ultra
log-concave random variables behaves on thinning. In the spirit of the
monotonicity results established by Artstein, Ball, Barthe and Naor, we prove a
stronger version of concavity of entropy, which implies a strengthened form of
our discrete Entropy Power Inequality.Comment: 9 pages (revised to take account of referees' comments
-interpolation of probability measures on graphs
We generalize an equation introduced by Benamou and Brenier in [BB00] and characterizing Wasserstein Wp-geodesics for p > 1, from the continuous setting of probability distributions on a Riemannian manifold to the discrete setting of probability distributions on a general graph. Given an initial and a final distributions (f0(x))x∈G, (f1(x))x∈G, we prove the existence of a curve (ft(k)) t∈[0,1],k∈Z satisfying this Benamou-Brenier equation. We also show that such a curve can be described as a mixture of binomial distributions with respect to a coupling that is solution of a certain optimization problem