217 research outputs found
Stationary quantum source coding
In this paper the quantum source coding theorem is obtained for a completely
ergodic source. This results extends Shannon's classical theorem as well as
Schumacher's quantum noiseless coding theorem for memoryless sources. The
control of the memory effects requires earlier results of Hiai and Petz on high
probability subspaces.Comment: 8 page
A nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution
We discuss a nonlinear model for the relaxation by energy redistribution
within an isolated, closed system composed of non-interacting identical
particles with energy levels e_i with i=1,2,...,N. The time-dependent
occupation probabilities p_i(t) are assumed to obey the nonlinear rate
equations tau dp_i/dt=-p_i ln p_i+ alpha(t)p_i-beta(t)e_ip_i where alpha(t) and
beta(t) are functionals of the p_i(t)'s that maintain invariant the mean energy
E=sum_i e_ip_i(t) and the normalization condition 1=sum_i p_i(t). The entropy
S(t)=-k sum_i p_i(t) ln p_i(t) is a non-decreasing function of time until the
initially nonzero occupation probabilities reach a Boltzmann-like canonical
distribution over the occupied energy eigenstates. Initially zero occupation
probabilities, instead, remain zero at all times. The solutions p_i(t) of the
rate equations are unique and well-defined for arbitrary initial conditions
p_i(0) and for all times. Existence and uniqueness both forward and backward in
time allows the reconstruction of the primordial lowest entropy state. The time
evolution is at all times along the local direction of steepest entropy ascent
or, equivalently, of maximal entropy generation. These rate equations have the
same mathematical structure and basic features of the nonlinear dynamical
equation proposed in a series of papers ended with G.P.Beretta, Found.Phys.,
17, 365 (1987) and recently rediscovered in S. Gheorghiu-Svirschevski,
Phys.Rev.A, 63, 022105 and 054102 (2001). Numerical results illustrate the
features of the dynamics and the differences with the rate equations recently
considered for the same problem in M.Lemanska and Z.Jaeger, Physica D, 170, 72
(2002).Comment: 11 pages, 7 eps figures (psfrag use removed), uses subeqn, minor
revisions, accepted for Physical Review
Fundamental properties of Tsallis relative entropy
Fundamental properties for the Tsallis relative entropy in both classical and
quantum systems are studied. As one of our main results, we give the parametric
extension of the trace inequality between the quantum relative entropy and the
minus of the trace of the relative operator entropy given by Hiai and Petz. The
monotonicity of the quantum Tsallis relative entropy for the trace preserving
completely positive linear map is also shown without the assumption that the
density operators are invertible.
The generalized Tsallis relative entropy is defined and its subadditivity is
shown by its joint convexity. Moreover, the generalized Peierls-Bogoliubov
inequality is also proven
On the monotonicity of scalar curvature in classical and quantum information geometry
We study the statistical monotonicity of the scalar curvature for the
alpha-geometries on the simplex of probability vectors. From the results
obtained and from numerical data we are led to some conjectures about quantum
alpha-geometries and Wigner-Yanase-Dyson information. Finally we show that this
last conjecture implies the truth of the Petz conjecture about the monotonicity
of the scalar curvature of the Bogoliubov-Kubo-Mori monotone metric.Comment: 20 pages, 2 .eps figures; (v2) section 2 rewritten, typos correcte
Maps on density operators preserving quantum f-divergences
For an arbitrary strictly convex function f defined on the
non-negative real line we determine the structure of all transformations
on the set of density operators which preserve the quantum f-divergence
Typical support and Sanov large deviations of correlated states
Discrete stationary classical processes as well as quantum lattice states are
asymptotically confined to their respective typical support, the exponential
growth rate of which is given by the (maximal ergodic) entropy. In the iid case
the distinguishability of typical supports can be asymptotically specified by
means of the relative entropy, according to Sanov's theorem. We give an
extension to the correlated case, referring to the newly introduced class of
HP-states.Comment: 29 pages, no figures, references adde
Canonical moments and random spectral measures
We study some connections between the random moment problem and the random
matrix theory. A uniform draw in a space of moments can be lifted into the
spectral probability measure of the pair (A,e) where A is a random matrix from
a classical ensemble and e is a fixed unit vector. This random measure is a
weighted sampling among the eigenvalues of A. We also study the large
deviations properties of this random measure when the dimension of the matrix
grows. The rate function for these large deviations involves the reversed
Kullback information.Comment: 32 pages. Revised version accepted for publication in Journal of
Theoretical Probabilit
Homogeneous Open Quantum Random Walks on a lattice
We study Open Quantum Random Walks for which the underlying graph is a
lattice, and the generators of the walk are translation-invariant. We consider
the quantum trajectory associated with the OQRW, which is described by a
position process and a state process. We obtain a central limit theorem and a
large deviation principle for the position process, and an ergodic result for
the state process. We study in detail the case of homogeneous OQRWs on a
lattice, with internal space
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