75 research outputs found
Quantum Brascamp-Lieb dualities
BrascampâLieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theoryâincluding entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of âgeometricâ BrascampâLieb inequalities
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
Mirror symmetry, Langlands duality, and the Hitchin system
We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or
equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact
Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in
two different senses. First, they satisfy the requirements laid down by
Strominger-Yau-Zaslow (SYZ), in a suitably general sense involving a B-field or
flat unitary gerbe. To show this, we use their hyperkahler structures and
Hitchin's integrable systems. Second, their Hodge numbers, again in a suitably
general sense, are equal. These spaces provide significant evidence in support
of SYZ. Moreover, they throw a bridge from mirror symmetry to the duality
theory of Lie groups and, more broadly, to the geometric Langlands program.Comment: 31 pages, LaTeX with packages amsfonts, latexsym, [dvips]graphicx,
[dvips]color, one embedded postscript figur
Transport Inequalities. A Survey
This is a survey of recent developments in the area of transport
inequalities. We investigate their consequences in terms of concentration and
deviation inequalities and sketch their links with other functional
inequalities and also large deviation theory.Comment: Proceedings of the conference Inhomogeneous Random Systems 2009; 82
pages
On contraction coefficients, partial orders and approximation of capacities for quantum channels
The data processing inequality is the most basic requirement for any
meaningful measure of information. It essentially states that
distinguishability measures between states decrease if we apply a quantum
channel. It is the centerpiece of many results in information theory and
justifies the operational interpretation of most entropic quantities. In this
work, we revisit the notion of contraction coefficients of quantum channels,
which provide sharper and specialized versions of the data processing
inequality. A concept closely related to data processing are partial orders on
quantum channels. We discuss several quantum extensions of the well known less
noisy ordering and then relate them to contraction coefficients. We further
define approximate versions of the partial orders and show how they can give
strengthened and conceptually simple proofs of several results on approximating
capacities. Moreover, we investigate the relation to other partial orders in
the literature and their properties, particularly with regards to
tensorization. We then investigate further properties of contraction
coefficients and their relation to other properties of quantum channels, such
as hypercontractivity. Next, we extend the framework of contraction
coefficients to general f-divergences and prove several structural results.
Finally, we consider two important classes of quantum channels, namely
Weyl-covariant and bosonic Gaussian channels. For those, we determine new
contraction coefficients and relations for various partial orders.Comment: 47 pages, 2 figure
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