117 research outputs found
Quantum Information on Spectral Sets
For convex optimization problems Bregman divergences appear as regret
functions. Such regret functions can be defined on any convex set but if a
sufficiency condition is added the regret function must be proportional to
information divergence and the convex set must be spectral. Spectral set are
sets where different orthogonal decompositions of a state into pure states have
unique mixing coefficients. Only on such spectral sets it is possible to define
well behaved information theoretic quantities like entropy and divergence. It
is only possible to perform measurements in a reversible way if the state space
is spectral. The most important spectral sets can be represented as positive
elements of Jordan algebras with trace 1. This means that Jordan algebras
provide a natural framework for studying quantum information. We compare
information theory on Hilbert spaces with information theory in more general
Jordan algebras, and conclude that much of the formalism is unchanged but also
identify some important differences.Comment: 13 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1701.0101
Lattices with non-Shannon Inequalities
We study the existence or absence of non-Shannon inequalities for variables
that are related by functional dependencies. Although the power-set on four
variables is the smallest Boolean lattice with non-Shannon inequalities there
exist lattices with many more variables without non-Shannon inequalities. We
search for conditions that ensures that no non-Shannon inequalities exist. It
is demonstrated that 3-dimensional distributive lattices cannot have
non-Shannon inequalities and planar modular lattices cannot have non-Shannon
inequalities. The existence of non-Shannon inequalities is related to the
question of whether a lattice is isomorphic to a lattice of subgroups of a
group.Comment: Ten pages. Submitted to ISIT 2015. The appendix will not appear in
the proceeding
Maximum Entropy and Sufficiency
The notion of Bregman divergence and sufficiency will be defined on general
convex state spaces. It is demonstrated that only spectral sets can have a
Bregman divergence that satisfies a sufficiency condition. Positive elements
with trace 1 in a Jordan algebra are examples of spectral sets, and the most
important example is the set of density matrices with complex entries. It is
conjectured that information theoretic considerations lead directly to the
notion of Jordan algebra under some regularity conditions
R\'enyi Divergence and Kullback-Leibler Divergence
R\'enyi divergence is related to R\'enyi entropy much like Kullback-Leibler
divergence is related to Shannon's entropy, and comes up in many settings. It
was introduced by R\'enyi as a measure of information that satisfies almost the
same axioms as Kullback-Leibler divergence, and depends on a parameter that is
called its order. In particular, the R\'enyi divergence of order 1 equals the
Kullback-Leibler divergence.
We review and extend the most important properties of R\'enyi divergence and
Kullback-Leibler divergence, including convexity, continuity, limits of
-algebras and the relation of the special order 0 to the Gaussian
dichotomy and contiguity. We also show how to generalize the Pythagorean
inequality to orders different from 1, and we extend the known equivalence
between channel capacity and minimax redundancy to continuous channel inputs
(for all orders) and present several other minimax results.Comment: To appear in IEEE Transactions on Information Theor
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