126 research outputs found
On the exact variance of Tsallis entropy in a random pure state
Tsallis entropy is a useful one-parameter generalization of the standard von
Neumann entropy in information theory. We study the variance of Tsallis entropy
of bipartite quantum systems in a random pure state. The main result is an
exact variance formula of Tsallis entropy that involves finite sums of some
terminating hypergeometric functions. In the special cases of quadratic entropy
and small subsystem dimensions, the main result is further simplified to
explicit variance expressions. As a byproduct, we find an independent proof of
the recently proved variance formula of von Neumann entropy based on the
derived moment relation to the Tsallis entropy
Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies
We introduce a new class of generalized isotropic Lipkin-Meshkov-Glick models
with su spin and long-range non-constant interactions, whose
non-degenerate ground state is a Dicke state of su type. We evaluate in
closed form the reduced density matrix of a block of spins when the whole
system is in its ground state, and study the corresponding von Neumann and
R\'enyi entanglement entropies in the thermodynamic limit. We show that both of
these entropies scale as when tends to infinity, where the
coefficient is equal to in the ground state phase with
vanishing su magnon densities. In particular, our results show that none
of these generalized Lipkin-Meshkov-Glick models are critical, since when
their R\'enyi entropy becomes independent of the parameter
. We have also computed the Tsallis entanglement entropy of the ground state
of these generalized su Lipkin-Meshkov-Glick models, finding that it can
be made extensive by an appropriate choice of its parameter only when
. Finally, in the su case we construct in detail the phase
diagram of the ground state in parameter space, showing that it is determined
in a simple way by the weights of the fundamental representation of su.
This is also true in the su case; for instance, we prove that the region
for which all the magnon densities are non-vanishing is an -simplex in
whose vertices are the weights of the fundamental representation
of su.Comment: Typeset with LaTeX, 32 pages, 3 figures. Final version with
corrections and additional reference
Entanglement, quantum randomness, and complexity beyond scrambling
Scrambling is a process by which the state of a quantum system is effectively
randomized due to the global entanglement that "hides" initially localized
quantum information. In this work, we lay the mathematical foundations of
studying randomness complexities beyond scrambling by entanglement properties.
We do so by analyzing the generalized (in particular R\'enyi) entanglement
entropies of designs, i.e. ensembles of unitary channels or pure states that
mimic the uniformly random distribution (given by the Haar measure) up to
certain moments. A main collective conclusion is that the R\'enyi entanglement
entropies averaged over designs of the same order are almost maximal. This
links the orders of entropy and design, and therefore suggests R\'enyi
entanglement entropies as diagnostics of the randomness complexity of
corresponding designs. Such complexities form a hierarchy between information
scrambling and Haar randomness. As a strong separation result, we prove the
existence of (state) 2-designs such that the R\'enyi entanglement entropies of
higher orders can be bounded away from the maximum. However, we also show that
the min entanglement entropy is maximized by designs of order only logarithmic
in the dimension of the system. In other words, logarithmic-designs already
achieve the complexity of Haar in terms of entanglement, which we also call
max-scrambling. This result leads to a generalization of the fast scrambling
conjecture, that max-scrambling can be achieved by physical dynamics in time
roughly linear in the number of degrees of freedom.Comment: 72 pages, 4 figures. Rewritten version with new title. v3: published
versio
Some Open Points in Nonextensive Statistical Mechanics
We present and discuss a list of some interesting points that are currently
open in nonextensive statistical mechanics. Their analytical, numerical,
experimental or observational advancement would naturally be very welcome.Comment: 30 pages including 6 figures. Invited paper to appear in the
International Journal of Bifurcation and Chao
Multipartite entanglement characterization of a quantum phase transition
A probability density characterization of multipartite entanglement is tested
on the one-dimensional quantum Ising model in a transverse field. The average
and second moment of the probability distribution are numerically shown to be
good indicators of the quantum phase transition. We comment on multipartite
entanglement generation at a quantum phase transition.Comment: 10 pages, 6 figures, final versio
Hilbert-Schmidt Separability Probabilities and Noninformativity of Priors
The Horodecki family employed the Jaynes maximum-entropy principle, fitting
the mean (b_{1}) of the Bell-CHSH observable (B). This model was extended by
Rajagopal by incorporating the dispersion (\sigma_{1}^2) of the observable, and
by Canosa and Rossignoli, by generalizing the observable (B_{\alpha}). We
further extend the Horodecki one-parameter model in both these manners,
obtaining a three-parameter (b_{1},\sigma_{1}^2,\alpha) two-qubit model, for
which we find a highly interesting/intricate continuum (-\infty < \alpha <
\infty) of Hilbert-Schmidt (HS) separability probabilities -- in which, the
golden ratio is featured. Our model can be contrasted with the three-parameter
(b_{q}, \sigma_{q}^2,q) one of Abe and Rajagopal, which employs a
q(Tsallis)-parameter rather than , and has simply q-invariant HS
separability probabilities of 1/2. Our results emerge in a study initially
focused on embedding certain information metrics over the two-level quantum
systems into a q-framework. We find evidence that Srednicki's recently-stated
biasedness criterion for noninformative priors yields rankings of priors fully
consistent with an information-theoretic test of Clarke, previously applied to
quantum systems by Slater.Comment: 26 pages, 12 figure
Entangled random pure states with orthogonal symmetry: exact results
We compute analytically the density of Schmidt
eigenvalues, distributed according to a fixed-trace Wishart-Laguerre measure,
and the average R\'enyi entropy for reduced
density matrices of entangled random pure states with orthogonal symmetry
. The results are valid for arbitrary dimensions of the
corresponding Hilbert space partitions, and are in excellent agreement with
numerical simulations.Comment: 15 pages, 5 figure
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