103 research outputs found
A volume inequality for quantum Fisher information and the uncertainty principle
Let be complex self-adjoint matrices and let be a
density matrix. The Robertson uncertainty principle gives a bound for the quantum
generalized covariance in terms of the commutators . The right side
matrix is antisymmetric and therefore the bound is trivial (equal to zero) in
the odd case .
Let be an arbitrary normalized symmetric operator monotone function and
let be the associated quantum Fisher information. In
this paper we conjecture the inequality that gives a
non-trivial bound for any natural number using the commutators . The inequality has been proved in the cases by the joint efforts
of many authors. In this paper we prove the case N=3 for real matrices
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
On the characterisation of paired monotone metrics
Hasegawa and Petz introduced the notion of dual statistically monotone
metrics. They also gave a characterisation theorem showing that
Wigner-Yanase-Dyson metrics are the only members of the dual family. In this
paper we show that the characterisation theorem holds true under more general
hypotheses.Comment: 12 pages, to appear on Ann. Inst. Stat. Math.; v2: changes made to
conform to accepted version, title changed as wel
Volume of the quantum mechanical state space
The volume of the quantum mechanical state space over -dimensional real,
complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean
measure is computed, and explicit formulas are presented for the expected value
of the determinant in the general setting too. The case when the state space is
endowed with a monotone metric or a pull-back metric is considered too, we give
formulas to compute the volume of the state space with respect to the given
Riemannian metric. We present the volume of the space of qubits with respect to
various monotone metrics. It turns out that the volume of the space of qubits
can be infinite too. We characterize those monotone metrics which generates
infinite volume.Comment: 17 page
Metric adjusted skew information: Convexity and restricted forms of superadditivity
We give a truly elementary proof of the convexity of metric adjusted skew
information following an idea of Effros. We extend earlier results of weak
forms of superadditivity to general metric adjusted skew informations.
Recently, Luo and Zhang introduced the notion of semi-quantum states on a
bipartite system and proved superadditivity of the Wigner-Yanase-Dyson skew
informations for such states. We extend this result to general metric adjusted
skew informations. We finally show that a recently introduced extension to
parameter values of the WYD-information is a special case of
(unbounded) metric adjusted skew information.Comment: An error in the literature is pointed ou
Evolution of a Primordial Black Hole Population
We reconsider in this work the effects of an energy absorption term in the
evolution of primordial black holes (hereafter PBHs) in the several epochs of
the Universe. A critical mass is introduced as a boundary between the accreting
and evaporating regimes of the PBHs. We show that the growth of PBHs is
negligible in the Radiation-dominated Era due to scarcity of energy density
supply from the expanding background, in agreement with a previous analysis by
Carr and Hawking, but that nevertheless the absorption term is large enough for
black holes above the critical mass to preclude their evaporation until the
universe has cooled sufficiently. The effects of PBH motion are also discussed:
the Doppler effect may give rise to energy accretion in black-holes with large
peculiar motions relative to background. We discuss how cosmological
constraints are modified by the introduction of the critical mass since that
PBHs above it do not disturb the CMBR. We show that there is a large range of
admissible masses for PBHs above the critical mass but well below the
cosmological horizon. Finally we outline a minimal kinetic formalism, solved in
some limiting cases, to deal with more complicated cases of PBH populationsComment: RevTex file, 8 pp., 3 .ps figures available upon request from
[email protected]
Nonparametric Information Geometry
The differential-geometric structure of the set of positive densities on a
given measure space has raised the interest of many mathematicians after the
discovery by C.R. Rao of the geometric meaning of the Fisher information. Most
of the research is focused on parametric statistical models. In series of
papers by author and coworkers a particular version of the nonparametric case
has been discussed. It consists of a minimalistic structure modeled according
the theory of exponential families: given a reference density other densities
are represented by the centered log likelihood which is an element of an Orlicz
space. This mappings give a system of charts of a Banach manifold. It has been
observed that, while the construction is natural, the practical applicability
is limited by the technical difficulty to deal with such a class of Banach
spaces. It has been suggested recently to replace the exponential function with
other functions with similar behavior but polynomial growth at infinity in
order to obtain more tractable Banach spaces, e.g. Hilbert spaces. We give
first a review of our theory with special emphasis on the specific issues of
the infinite dimensional setting. In a second part we discuss two specific
topics, differential equations and the metric connection. The position of this
line of research with respect to other approaches is briefly discussed.Comment: Submitted for publication in the Proceedings od GSI2013 Aug 28-30
2013 Pari
Information geometry and sufficient statistics
Information geometry provides a geometric approach to families of statistical
models. The key geometric structures are the Fisher quadratic form and the
Amari-Chentsov tensor. In statistics, the notion of sufficient statistic
expresses the criterion for passing from one model to another without loss of
information. This leads to the question how the geometric structures behave
under such sufficient statistics. While this is well studied in the finite
sample size case, in the infinite case, we encounter technical problems
concerning the appropriate topologies. Here, we introduce notions of
parametrized measure models and tensor fields on them that exhibit the right
behavior under statistical transformations. Within this framework, we can then
handle the topological issues and show that the Fisher metric and the
Amari-Chentsov tensor on statistical models in the class of symmetric 2-tensor
fields and 3-tensor fields can be uniquely (up to a constant) characterized by
their invariance under sufficient statistics, thereby achieving a full
generalization of the original result of Chentsov to infinite sample sizes.
More generally, we decompose Markov morphisms between statistical models in
terms of statistics. In particular, a monotonicity result for the Fisher
information naturally follows.Comment: 37 p, final version, minor corrections, improved presentatio
A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space
We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on Rd. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports the Fisher-Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this "lifts" to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space W²'¹ i.e. it maps each of these spaces continuously into itself. It also maps continuously between other fixed-norm spaces with a loss of Lebesgue exponent that increases with the number of derivatives. Some of the results make essential use of a log-Sobolev embedding theorem. Each manifold contains a smoothly embedded submanifold of probability measures. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker-Planck equation) are outlined
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