2,096 research outputs found
On Variational Expressions for Quantum Relative Entropies
Distance measures between quantum states like the trace distance and the
fidelity can naturally be defined by optimizing a classical distance measure
over all measurement statistics that can be obtained from the respective
quantum states. In contrast, Petz showed that the measured relative entropy,
defined as a maximization of the Kullback-Leibler divergence over projective
measurement statistics, is strictly smaller than Umegaki's quantum relative
entropy whenever the states do not commute. We extend this result in two ways.
First, we show that Petz' conclusion remains true if we allow general positive
operator valued measures. Second, we extend the result to Renyi relative
entropies and show that for non-commuting states the sandwiched Renyi relative
entropy is strictly larger than the measured Renyi relative entropy for , and strictly smaller for . The
latter statement provides counterexamples for the data-processing inequality of
the sandwiched Renyi relative entropy for . Our main tool is
a new variational expression for the measured Renyi relative entropy, which we
further exploit to show that certain lower bounds on quantum conditional mutual
information are superadditive.Comment: v2: final published versio
A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
We consider two fundamental tasks in quantum information theory, data
compression with quantum side information as well as randomness extraction
against quantum side information. We characterize these tasks for general
sources using so-called one-shot entropies. We show that these
characterizations - in contrast to earlier results - enable us to derive tight
second order asymptotics for these tasks in the i.i.d. limit. More generally,
our derivation establishes a hierarchy of information quantities that can be
used to investigate information theoretic tasks in the quantum domain: The
one-shot entropies most accurately describe an operational quantity, yet they
tend to be difficult to calculate for large systems. We show that they
asymptotically agree up to logarithmic terms with entropies related to the
quantum and classical information spectrum, which are easier to calculate in
the i.i.d. limit. Our techniques also naturally yields bounds on operational
quantities for finite block lengths.Comment: See also arXiv:1208.1400, which independently derives part of our
result: the second order asymptotics for binary hypothesis testin
An axiomatic characterization of a two-parameter extended relative entropy
The uniqueness theorem for a two-parameter extended relative entropy is
proven. This result extends our previous one, the uniqueness theorem for a
one-parameter extended relative entropy, to a two-parameter case. In addition,
the properties of a two-parameter extended relative entropy are studied.Comment: 11 page
Decompositions of factor codes and embeddings between shift spaces with unequal entropies
Given a factor code between sofic shifts X and Y, there is a family of
decompositions of the original code into factor codes such that the entropies
of the intermediate subshifts arising from the decompositions are dense in the
interval from the entropy of Y to that of X. Furthermore, if X is of finite
type, we can choose those intermediate subshifts as shifts of finite type. In
the second part of the paper, given an embedding from a shift space to an
irreducible sofic shift, we characterize the set of the entropies of the
intermediate subshifts arising from the decompositions of the given embedding
into embeddings.Comment: 14pages, 2 figures; v2) minor revision. to appear in Ergodic Theory
Dynamical System
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