4,787 research outputs found
Geometric inequalities from phase space translations
We establish a quantum version of the classical isoperimetric inequality
relating the Fisher information and the entropy power of a quantum state. The
key tool is a Fisher information inequality for a state which results from a
certain convolution operation: the latter maps a classical probability
distribution on phase space and a quantum state to a quantum state. We show
that this inequality also gives rise to several related inequalities whose
counterparts are well-known in the classical setting: in particular, it implies
an entropy power inequality for the mentioned convolution operation as well as
the isoperimetric inequality, and establishes concavity of the entropy power
along trajectories of the quantum heat diffusion semigroup. As an application,
we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck
semigroup, and argue that it implies fast convergence towards the fixed point
for a large class of initial states.Comment: 37 pages; updated to match published versio
Information Theoretic Proofs of Entropy Power Inequalities
While most useful information theoretic inequalities can be deduced from the
basic properties of entropy or mutual information, up to now Shannon's entropy
power inequality (EPI) is an exception: Existing information theoretic proofs
of the EPI hinge on representations of differential entropy using either Fisher
information or minimum mean-square error (MMSE), which are derived from de
Bruijn's identity. In this paper, we first present an unified view of these
proofs, showing that they share two essential ingredients: 1) a data processing
argument applied to a covariance-preserving linear transformation; 2) an
integration over a path of a continuous Gaussian perturbation. Using these
ingredients, we develop a new and brief proof of the EPI through a mutual
information inequality, which replaces Stam and Blachman's Fisher information
inequality (FII) and an inequality for MMSE by Guo, Shamai and Verd\'u used in
earlier proofs. The result has the advantage of being very simple in that it
relies only on the basic properties of mutual information. These ideas are then
generalized to various extended versions of the EPI: Zamir and Feder's
generalized EPI for linear transformations of the random variables, Takano and
Johnson's EPI for dependent variables, Liu and Viswanath's
covariance-constrained EPI, and Costa's concavity inequality for the entropy
power.Comment: submitted for publication in the IEEE Transactions on Information
Theory, revised versio
Brascamp-Lieb Inequality and Its Reverse: An Information Theoretic View
We generalize a result by Carlen and Cordero-Erausquin on the equivalence
between the Brascamp-Lieb inequality and the subadditivity of relative entropy
by allowing for random transformations (a broadcast channel). This leads to a
unified perspective on several functional inequalities that have been gaining
popularity in the context of proving impossibility results. We demonstrate that
the information theoretic dual of the Brascamp-Lieb inequality is a convenient
setting for proving properties such as data processing, tensorization,
convexity and Gaussian optimality. Consequences of the latter include an
extension of the Brascamp-Lieb inequality allowing for Gaussian random
transformations, the determination of the multivariate Wyner common information
for Gaussian sources, and a multivariate version of Nelson's hypercontractivity
theorem. Finally we present an information theoretic characterization of a
reverse Brascamp-Lieb inequality involving a random transformation (a multiple
access channel).Comment: 5 pages; to be presented at ISIT 201
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