1,361 research outputs found
The structure of Renyi entropic inequalities
We investigate the universal inequalities relating the alpha-Renyi entropies
of the marginals of a multi-partite quantum state. This is in analogy to the
same question for the Shannon and von Neumann entropy (alpha=1) which are known
to satisfy several non-trivial inequalities such as strong subadditivity.
Somewhat surprisingly, we find for 0<alpha<1, that the only inequality is
non-negativity: In other words, any collection of non-negative numbers assigned
to the nonempty subsets of n parties can be arbitrarily well approximated by
the alpha-entropies of the 2^n-1 marginals of a quantum state.
For alpha>1 we show analogously that there are no non-trivial homogeneous (in
particular no linear) inequalities. On the other hand, it is known that there
are further, non-linear and indeed non-homogeneous, inequalities delimiting the
alpha-entropies of a general quantum state.
Finally, we also treat the case of Renyi entropies restricted to classical
states (i.e. probability distributions), which in addition to non-negativity
are also subject to monotonicity. For alpha different from 0 and 1 we show that
this is the only other homogeneous relation.Comment: 15 pages. v2: minor technical changes in Theorems 10 and 1
Information geometric methods for complexity
Research on the use of information geometry (IG) in modern physics has
witnessed significant advances recently. In this review article, we report on
the utilization of IG methods to define measures of complexity in both
classical and, whenever available, quantum physical settings. A paradigmatic
example of a dramatic change in complexity is given by phase transitions (PTs).
Hence we review both global and local aspects of PTs described in terms of the
scalar curvature of the parameter manifold and the components of the metric
tensor, respectively. We also report on the behavior of geodesic paths on the
parameter manifold used to gain insight into the dynamics of PTs. Going
further, we survey measures of complexity arising in the geometric framework.
In particular, we quantify complexity of networks in terms of the Riemannian
volume of the parameter space of a statistical manifold associated with a given
network. We are also concerned with complexity measures that account for the
interactions of a given number of parts of a system that cannot be described in
terms of a smaller number of parts of the system. Finally, we investigate
complexity measures of entropic motion on curved statistical manifolds that
arise from a probabilistic description of physical systems in the presence of
limited information. The Kullback-Leibler divergence, the distance to an
exponential family and volumes of curved parameter manifolds, are examples of
essential IG notions exploited in our discussion of complexity. We conclude by
discussing strengths, limits, and possible future applications of IG methods to
the physics of complexity.Comment: review article, 60 pages, no figure
Remarks on entanglement entropy for gauge fields
In gauge theories the presence of constraints can obstruct expressing the
global Hilbert space as a tensor product of the Hilbert spaces corresponding to
degrees of freedom localized in complementary regions. In algebraic terms, this
is due to the presence of a center --- a set of operators which commute with
all others --- in the gauge invariant operator algebra corresponding to finite
region. A unique entropy can be assigned to algebras with center, giving place
to a local entropy in lattice gauge theories. However, ambiguities arise on the
correspondence between algebras and regions. In particular, it is always
possible to choose (in many different ways) local algebras with trivial center,
and hence a genuine entanglement entropy, for any region. These choices are in
correspondence with maximal trees of links on the boundary, which can be
interpreted as partial gauge fixings. This interpretation entails a gauge
fixing dependence of the entanglement entropy. In the continuum limit however,
ambiguities in the entropy are given by terms local on the boundary of the
region, in such a way relative entropy and mutual information are finite,
universal, and gauge independent quantities.Comment: 26 pages, 7 figure
A survey of uncertainty principles and some signal processing applications
The goal of this paper is to review the main trends in the domain of
uncertainty principles and localization, emphasize their mutual connections and
investigate practical consequences. The discussion is strongly oriented
towards, and motivated by signal processing problems, from which significant
advances have been made recently. Relations with sparse approximation and
coding problems are emphasized
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