17,264 research outputs found
A Continuity Theorem for Stinespring's Dilation
We show a continuity theorem for Stinespring's dilation: two completely
positive maps between arbitrary C*-algebras are close in cb-norm iff we can
find corresponding dilations that are close in operator norm. The proof
establishes the equivalence of the cb-norm distance and the Bures distance for
completely positive maps. We briefly discuss applications to quantum
information theory.Comment: 18 pages, no figure
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
Continuity of the Maximum-Entropy Inference
We study the inverse problem of inferring the state of a finite-level quantum
system from expected values of a fixed set of observables, by maximizing a
continuous ranking function. We have proved earlier that the maximum-entropy
inference can be a discontinuous map from the convex set of expected values to
the convex set of states because the image contains states of reduced support,
while this map restricts to a smooth parametrization of a Gibbsian family of
fully supported states. Here we prove for arbitrary ranking functions that the
inference is continuous up to boundary points. This follows from a continuity
condition in terms of the openness of the restricted linear map from states to
their expected values. The openness condition shows also that ranking functions
with a discontinuous inference are typical. Moreover it shows that the
inference is continuous in the restriction to any polytope which implies that a
discontinuity belongs to the quantum domain of non-commutative observables and
that a geodesic closure of a Gibbsian family equals the set of maximum-entropy
states. We discuss eight descriptions of the set of maximum-entropy states with
proofs of accuracy and an analysis of deviations.Comment: 34 pages, 1 figur
An Algebraic Characterization of Vacuum States in Minkowski Space. III. Reflection Maps
Employing the algebraic framework of local quantum physics, vacuum states in
Minkowski space are distinguished by a property of geometric modular action.
This property allows one to construct from any locally generated net of
observables and corresponding state a continuous unitary representation of the
proper Poincare group which acts covariantly on the net and leaves the state
invariant. The present results and methods substantially improve upon previous
work. In particular, the continuity properties of the representation are shown
to be a consequence of the net structure, and surmised cohomological problems
in the construction of the representation are resolved by demonstrating that,
for the Poincare group, continuous reflection maps are restrictions of
continuous homomorphisms.Comment: 20 pages; change of title, reference added; version as to appear in
Commun. Math. Phy
Ergodic Classical-Quantum Channels: Structure and Coding Theorems
We consider ergodic causal classical-quantum channels (cq-channels) which
additionally have a decaying input memory. In the first part we develop some
structural properties of ergodic cq-channels and provide equivalent conditions
for ergodicity. In the second part we prove the coding theorem with weak
converse for causal ergodic cq-channels with decaying input memory. Our proof
is based on the possibility to introduce joint input-output state for the
cq-channels and an application of the Shannon-McMillan theorem for ergodic
quantum states. In the last part of the paper it is shown how this result
implies coding theorem for the classical capacity of a class of causal ergodic
quantum channels.Comment: 19 pages, no figures. Final versio
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