12,267 research outputs found
Entropy in general physical theories
Information plays an important role in our understanding of the physical
world. We hence propose an entropic measure of information for any physical
theory that admits systems, states and measurements. In the quantum and
classical world, our measure reduces to the von Neumann and Shannon entropy
respectively. It can even be used in a quantum or classical setting where we
are only allowed to perform a limited set of operations. In a world that admits
superstrong correlations in the form of non-local boxes, our measure can be
used to analyze protocols such as superstrong random access encodings and the
violation of `information causality'. However, we also show that in such a
world no entropic measure can exhibit all properties we commonly accept in a
quantum setting. For example, there exists no`reasonable' measure of
conditional entropy that is subadditive. Finally, we prove a coding theorem for
some theories that is analogous to the quantum and classical setting, providing
us with an appealing operational interpretation.Comment: 20 pages, revtex, 7 figures, v2: Coding theorem revised, published
versio
Sampling of min-entropy relative to quantum knowledge
Let X_1, ..., X_n be a sequence of n classical random variables and consider
a sample of r positions selected at random. Then, except with (exponentially in
r) small probability, the min-entropy of the sample is not smaller than,
roughly, a fraction r/n of the total min-entropy of all positions X_1, ...,
X_n, which is optimal. Here, we show that this statement, originally proven by
Vadhan [LNCS, vol. 2729, Springer, 2003] for the purely classical case, is
still true if the min-entropy is measured relative to a quantum system. Because
min-entropy quantifies the amount of randomness that can be extracted from a
given random variable, our result can be used to prove the soundness of locally
computable extractors in a context where side information might be
quantum-mechanical. In particular, it implies that key agreement in the
bounded-storage model (using a standard sample-and-hash protocol) is fully
secure against quantum adversaries, thus solving a long-standing open problem.Comment: 48 pages, late
Mutual and coherent informations for infinite-dimensional quantum channels
The work is devoted to study of quantum mutual information and coherent
information -- the two important characteristics of quantum communication
channel. Appropriate definitions of these quantities in the
infinite-dimensional case are given and their properties are studied in detail.
Basic identities relating quantum mutual information and coherent information
of a pair of complementary channels are proved. An unexpected continuity
property of quantum mutual information and coherent information following from
the above identities is observed. An upper bound for the coherent information
is obtained.Comment: 27 pages, an alternative expression for the coherent information is
added (Remark 2
Generalized Entropies
We study an entropy measure for quantum systems that generalizes the von
Neumann entropy as well as its classical counterpart, the Gibbs or Shannon
entropy. The entropy measure is based on hypothesis testing and has an elegant
formulation as a semidefinite program, a type of convex optimization. After
establishing a few basic properties, we prove upper and lower bounds in terms
of the smooth entropies, a family of entropy measures that is used to
characterize a wide range of operational quantities. From the formulation as a
semidefinite program, we also prove a result on decomposition of hypothesis
tests, which leads to a chain rule for the entropy.Comment: 21 page
Tests for quantum contextuality in terms of -entropies
The information-theoretic approach to Bell's theorem is developed with use of
the conditional -entropies. The -entropic measures fulfill many similar
properties to the standard Shannon entropy. In general, both the locality and
noncontextuality notions are usually treated with use of the so-called marginal
scenarios. These hypotheses lead to the existence of a joint probability
distribution, which marginalizes to all particular ones. Assuming the existence
of such a joint probability distribution, we derive the family of inequalities
of Bell's type in terms of conditional -entropies for all . Quantum
violations of the new inequalities are exemplified within the
Clauser-Horne-Shimony-Holt (CHSH) and Klyachko-Can-Binicio\v{g}lu-Shumovsky
(KCBS) scenarios. An extension to the case of -cycle scenario is briefly
mentioned. The new inequalities with conditional -entropies allow to expand
a class of probability distributions, for which the nonlocality or
contextuality can be detected within entropic formulation. The -entropic
inequalities can also be useful in analyzing cases with detection
inefficiencies. Using two models of such a kind, we consider some potential
advantages of the -entropic formulation.Comment: 14 pages, two figures. The version 3 matches the journal versio
Quantum data compression, quantum information generation, and the density-matrix renormalization group method
We have studied quantum data compression for finite quantum systems where the
site density matrices are not independent, i.e., the density matrix cannot be
given as direct product of site density matrices and the von Neumann entropy is
not equal to the sum of site entropies. Using the density-matrix
renormalization group (DMRG) method for the 1-d Hubbard model, we have shown
that a simple relationship exists between the entropy of the left or right
block and dimension of the Hilbert space of that block as well as of the
superblock for any fixed accuracy. The information loss during the RG procedure
has been investigated and a more rigorous control of the relative error has
been proposed based on Kholevo's theory. Our results are also supported by the
quantum chemistry version of DMRG applied to various molecules with system
lengths up to 60 lattice sites. A sum rule which relates site entropies and the
total information generated by the renormalization procedure has also been
given which serves as an alternative test of convergence of the DMRG method.Comment: 8 pages, 7 figure
Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
We study dynamical optimal transport metrics between density matrices
associated to symmetric Dirichlet forms on finite-dimensional -algebras.
Our setting covers arbitrary skew-derivations and it provides a unified
framework that simultaneously generalizes recently constructed transport
metrics for Markov chains, Lindblad equations, and the Fermi
Ornstein--Uhlenbeck semigroup. We develop a non-nommutative differential
calculus that allows us to obtain non-commutative Ricci curvature bounds,
logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral
gap estimates
R\'enyi and Tsallis formulations of noise-disturbance trade-off relations
We address an information-theoretic approach to noise and disturbance in
quantum measurements. Properties of corresponding probability distributions are
characterized by means of both the R\'{e}nyi and Tsallis entropies. Related
information-theoretic measures of noise and disturbance are introduced. These
definitions are based on the concept of conditional entropy. To motivate
introduced measures, some important properties of the conditional R\'{e}nyi and
Tsallis entropies are discussed. There exist several formulations of entropic
uncertainty relations for a pair of observables. Trade-off relations for noise
and disturbance are derived on the base of known formulations of such a kind.Comment: 14 pages, no figures. The version 3 corresponds to the journal
versio
On quantum Renyi entropies: a new generalization and some properties
The Renyi entropies constitute a family of information measures that
generalizes the well-known Shannon entropy, inheriting many of its properties.
They appear in the form of unconditional and conditional entropies, relative
entropies or mutual information, and have found many applications in
information theory and beyond. Various generalizations of Renyi entropies to
the quantum setting have been proposed, most notably Petz's quasi-entropies and
Renner's conditional min-, max- and collision entropy. Here, we argue that
previous quantum extensions are incompatible and thus unsatisfactory.
We propose a new quantum generalization of the family of Renyi entropies that
contains the von Neumann entropy, min-entropy, collision entropy and the
max-entropy as special cases, thus encompassing most quantum entropies in use
today. We show several natural properties for this definition, including
data-processing inequalities, a duality relation, and an entropic uncertainty
relation.Comment: v1: contains several conjectures; v2: conjectures are resolved - see
also arXiv:1306.5358 and arXiv:1306.5920; v3: published versio
Towards an Entanglement Measure for Mixed States in CFTs Based on Relative Entropy
Relative entropy of entanglement (REE) is an entanglement measure of
bipartite mixed states, defined by the minimum of the relative entropy
between a given mixed state and an
arbitrary separable state . The REE is always bounded by the
mutual information because
the latter measures not only quantum entanglement but also classical
correlations. In this paper we address the question of to what extent REE can
be small compared to the mutual information in conformal field theories (CFTs).
For this purpose, we perturbatively compute the relative entropy between the
vacuum reduced density matrix on disjoint subsystems
and arbitrarily separable state in the limit where two subsystems
A and B are well separated, then minimize the relative entropy with respect to
the separable states. We argue that the result highly depends on the spectrum
of CFT on the subsystems. When we have a few low energy spectrum of operators
as in the case where the subsystems consist of a finite number of spins in spin
chain models, the REE is considerably smaller than the mutual information.
However in general our perturbative scheme breaks down, and the REE can be as
large as the mutual information.Comment: 35 pages, 2 figure
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