1,693 research outputs found
Comment on "Towards a large deviation theory for strongly correlated systems"
I comment on a recent paper by Ruiz and Tsallis [Phys. Lett. A 376, 2451
(2012)] claiming to have found a '-exponential' generalization of the large
deviation principle for strongly correlated random variables. I show that the
basic scaling results that they find numerically can be reproduced with a
simple example involving independent random variables, and are not specifically
related to the -exponential function. In fact, identical scaling results can
be obtained with any other power-law deformations of the exponential. Thus
their results do not conclusively support their claim of a -exponential
generalization of the large deviation principle.Comment: Comment, 3 pages, 2 figure
Introduction to dynamical large deviations of Markov processes
These notes give a summary of techniques used in large deviation theory to
study the fluctuations of time-additive quantities, called dynamical
observables, defined in the context of Langevin-type equations, which model
equilibrium and nonequilibrium processes driven by external forces and noise
sources. These fluctuations are described by large deviation functions,
obtained by solving a dominant eigenvalue problem similar to the problem of
finding the ground state energy of quantum systems. This analogy is used to
explain the differences that exist between the fluctuations of equilibrium and
nonequilibrium processes. An example involving the Ornstein-Uhlenbeck process
is worked out in detail to illustrate these methods. Exercises, at the end of
the notes, also complement the theory.Comment: 19 pages. Lecture notes for the 2017 Summer School on Fundamental
Problems in Statistical Physics XIV, 16-29 July 2017, Bruneck (Brunico),
Italy. v2: Typos corrected, exercises added. v3: Typos corrected. v4: More
typos corrected, footnote and references added. v5: Close to published
version. I dedicate this paper to the memory of E. G. D. Cohen (1923-2017
Asymptotic equivalence of probability measures and stochastic processes
Let and be two probability measures representing two different
probabilistic models of some system (e.g., an -particle equilibrium system,
a set of random graphs with vertices, or a stochastic process evolving over
a time ) and let be a random variable representing a 'macrostate' or
'global observable' of that system. We provide sufficient conditions, based on
the Radon-Nikodym derivative of and , for the set of typical values
of obtained relative to to be the same as the set of typical values
obtained relative to in the limit . This extends to
general probability measures and stochastic processes the well-known
thermodynamic-limit equivalence of the microcanonical and canonical ensembles,
related mathematically to the asymptotic equivalence of conditional and
exponentially-tilted measures. In this more general sense, two probability
measures that are asymptotically equivalent predict the same typical or
macroscopic properties of the system they are meant to model.Comment: v1: 16 pages. v2: 17 pages, precisions, examples and references
added. v3: Minor typos corrected. Close to published versio
Quantum Information Complexity and Amortized Communication
We define a new notion of information cost for quantum protocols, and a
corresponding notion of quantum information complexity for bipartite quantum
channels, and then investigate the properties of such quantities. These are the
fully quantum generalizations of the analogous quantities for bipartite
classical functions that have found many applications recently, in particular
for proving communication complexity lower bounds. Our definition is strongly
tied to the quantum state redistribution task.
Previous attempts have been made to define such a quantity for quantum
protocols, with particular applications in mind; our notion differs from these
in many respects. First, it directly provides a lower bound on the quantum
communication cost, independent of the number of rounds of the underlying
protocol. Secondly, we provide an operational interpretation for quantum
information complexity: we show that it is exactly equal to the amortized
quantum communication complexity of a bipartite channel on a given state. This
generalizes a result of Braverman and Rao to quantum protocols, and even
strengthens the classical result in a bounded round scenario. Also, this
provides an analogue of the Schumacher source compression theorem for
interactive quantum protocols, and answers a question raised by Braverman.
We also discuss some potential applications to quantum communication
complexity lower bounds by specializing our definition for classical functions
and inputs. Building on work of Jain, Radhakrishnan and Sen, we provide new
evidence suggesting that the bounded round quantum communication complexity of
the disjointness function is \Omega (n/M + M), for M-message protocols. This
would match the best known upper bound.Comment: v1, 38 pages, 1 figur
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