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 $n$-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

Geodesic distances on density matrices

We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.Comment: 10 page

Inequalities for quantum skew information

We study quantum information inequalities and show that the basic inequality between the quantum variance and the metric adjusted skew information generates all the multi-operator matrix inequalities or Robertson type determinant inequalities studied by a number of authors. We introduce an order relation on the set of functions representing quantum Fisher information that renders the set into a lattice with an involution. This order structure generates new inequalities for the metric adjusted skew informations. In particular, the Wigner-Yanase skew information is the maximal skew information with respect to this order structure in the set of Wigner-Yanase-Dyson skew informations. Key words and phrases: Quantum covariance, metric adjusted skew information, Robertson-type uncertainty principle, operator monotone function, Wigner-Yanase-Dyson skew information

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 $1<p\le 2$ of the WYD-information is a special case of (unbounded) metric adjusted skew information.Comment: An error in the literature is pointed ou

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

Pattern recognition receptors as potential therapeutic targets in inflammatory rheumatic disease

The pattern recognition receptors of the innate immune system are part of the first line of defence against pathogens. However, they also have the ability to respond to danger signals that are frequently elevated during tissue damage and at sites of inflammation. Inadvertent activation of pattern recognition receptors has been proposed to contribute to the pathogenesis of many conditions including inflammatory rheumatic diseases. Prolonged inflammation most often results in pain and damage to tissues. In particular, the Toll-like receptors and nucleotide-binding oligomerisation domain-like receptors that form inflammasomes have been postulated as key contributors to the inflammation observed in rheumatoid arthritis, osteoarthritis, gout and systemic lupus erythematosus. As such, there is increasing interest in targeting these receptors for therapeutic treatment in the clinic. Here the role of pattern recognition receptors in the pathogenesis of these diseases is discussed, with an update on the development of interventions to modulate the activity of these potential therapeutic targets