An inequality for expectation of means of positive random variables

Suppose that $X,Y$ are positive random variable and $m$ a numerical (commutative) mean. We prove that the inequality ${\rm E} (m(X,Y)) \leq m({\rm E} (X), {\rm E} (Y))$ holds if and only if the mean is generated by a concave function. With due changes we also prove that the same inequality holds for all operator means in the Kubo-Ando setting. The case of the harmonic mean was proved by C.R. Rao and B.L.S. Prakasa Rao

A Robertson-type Uncertainty Principle and Quantum Fisher Information

Let $A_1,...,A_N$ be complex selfadjoint matrices and let $\rho$ be a density matrix. The Robertson uncertainty principle $$det (Cov_\rho(A_h,A_j)) \geq det (- \frac{i}{2} Tr (\rho [A_h,A_j]))$$ gives a bound for the quantum generalized covariance in terms of the commutators $[A_h,A_j]$. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case $N=2m+1$. Let $f$ be an arbitrary normalized symmetric operator monotone function and let $_{\rho,f}$ be the associated quantum Fisher information. In this paper we prove the inequality $$det (Cov_\rho (A_h,A_j)) \geq det (\frac{f(0)}{2} _{\rho,f})$$ that gives a non-trivial bound for any $N \in {\mathbb N}$ using the commutators $[\rho,A_h]$.Comment: 17 pages (approx.

Uncertainty principle for Wigner-Yanase-Dyson information in semifinite von Neumann algebras

Recently Kosaki proved an uncertainty principle for matrices, related to Wigner-Yanase-Dyson information, and asked if a similar inequality could be proved in the von Neumann algebra setting. In this paper we prove such an uncertainty principle in the semifinite case

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

A unified approach to Local Quantum Uncertainty and Interferometric Power by Metric Adjusted Skew Information

Local quantum uncertainty and interferometric power have been introduced by Girolami et al. in [1,2] as geometric quantifiers of quantum correlations. The aim of the present paper is to discuss their properties in a unified manner by means of the the metric adjusted skew information defined by Hansen in [3].Comment: submitted to Entrop

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