61,919 research outputs found
Entropy accumulation
We ask the question whether entropy accumulates, in the sense that the
operationally relevant total uncertainty about an -partite system corresponds to the sum of the entropies of its parts . The
Asymptotic Equipartition Property implies that this is indeed the case to first
order in , under the assumption that the parts are identical and
independent of each other. Here we show that entropy accumulation occurs more
generally, i.e., without an independence assumption, provided one quantifies
the uncertainty about the individual systems by the von Neumann entropy
of suitably chosen conditional states. The analysis of a large system can hence
be reduced to the study of its parts. This is relevant for applications. In
device-independent cryptography, for instance, the approach yields essentially
optimal security bounds valid for general attacks, as shown by Arnon-Friedman
et al.Comment: 44 pages; expandable to 48 page
Orders of accumulation of entropy
For a continuous map of a compact metrizable space with finite
topological entropy, the order of accumulation of entropy of is a countable
ordinal that arises in the context of entropy structure and symbolic
extensions. We show that every countable ordinal is realized as the order of
accumulation of some dynamical system. Our proof relies on functional analysis
of metrizable Choquet simplices and a realization theorem of Downarowicz and
Serafin. Further, if is a metrizable Choquet simplex, we bound the ordinals
that appear as the order of accumulation of entropy of a dynamical system whose
simplex of invariant measures is affinely homeomorphic to . These bounds are
given in terms of the Cantor-Bendixson rank of \overline{\ex(M)}, the closure
of the extreme points of , and the relative Cantor-Bendixson rank of
\overline{\ex(M)} with respect to \ex(M). We also address the optimality of
these bounds.Comment: 48 page
Renormalized entropy for one dimensional discrete maps: periodic and quasi-periodic route to chaos and their robustness
We apply renormalized entropy as a complexity measure to the logistic and
sine-circle maps. In the case of logistic map, renormalized entropy decreases
(increases) until the accumulation point (after the accumulation point up to
the most chaotic state) as a sign of increasing (decreasing) degree of order in
all the investigated periodic windows, namely, period-2, 3, and 5, thereby
proving the robustness of this complexity measure. This observed change in the
renormalized entropy is adequate, since the bifurcations are exhibited before
the accumulation point, after which the band-merging, in opposition to the
bifurcations, is exhibited. In addition to the precise detection of the
accumulation points in all these windows, it is shown that the renormalized
entropy can detect the self-similar windows in the chaotic regime by exhibiting
abrupt changes in its values. Regarding the sine-circle map, we observe that
the renormalized entropy detects also the quasi-periodic regimes by showing
oscillatory behavior particularly in these regimes. Moreover, the oscillatory
regime of the renormalized entropy corresponds to a larger interval of the
nonlinearity parameter of the sine-circle map as the value of the frequency
ratio parameter reaches the critical value, at which the winding ratio attains
the golden mean.Comment: 14 pages, 7 figure
Generalised entropy accumulation
Consider a sequential process in which each step outputs a system and
updates a side information register . We prove that if this process
satisfies a natural "non-signalling" condition between past outputs and future
side information, the min-entropy of the outputs conditioned
on the side information at the end of the process can be bounded from below
by a sum of von Neumann entropies associated with the individual steps. This is
a generalisation of the entropy accumulation theorem (EAT), which deals with a
more restrictive model of side information: there, past side information cannot
be updated in subsequent rounds, and newly generated side information has to
satisfy a Markov condition. Due to its more general model of side-information,
our generalised EAT can be applied more easily and to a broader range of
cryptographic protocols. As examples, we give the first multi-round security
proof for blind randomness expansion and a simplified analysis of the E91 QKD
protocol. The proof of our generalised EAT relies on a new variant of Uhlmann's
theorem and new chain rules for the Renyi divergence and entropy, which might
be of independent interest.Comment: 42 pages; v2 expands introduction but does not change any results; in
FOCS 202
Sensitivity function and entropy increase rates for z-logistic map family at the edge of chaos
It is well known that, for chaotic systems, the production of relevant
entropy (Boltzmann-Gibbs) is always linear and the system has strong
(exponential) sensitivity to initial conditions. In recent years, various
numerical results indicate that basically the same type of behavior emerges at
the edge of chaos if a specific generalization of the entropy and the
exponential are used. In this work, we contribute to this scenario by
numerically analysing some generalized nonextensive entropies and their related
exponential definitions using -logistic map family. We also corroborate our
findings by testing them at accumulation points of different cycles.Comment: 9 pages, 2 fig
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