Wigner-Yanase information on quantum state space:the geometric approach

In the search of appropriate riemannian metrics on quantum state space the concept of statistical monotonicity, or contraction under coarse graining, has been proposed by Chentsov. The metrics with this property have been classified by Petz. All the elements of this family of geometries can be seen as quantum analogues of Fisher information. Although there exists a number of general theorems sheding light on this subject, many natural questions, also stemming from applications, are still open. In this paper we discuss a particular member of the family, the Wigner-Yanase information. Using a well-known approach that mimics the classical pull-back approach to Fisher information, we are able to give explicit formulae for the geodesic distance, the geodesic path, the sectional and scalar curvatures associated to Wigner-Yanase information. Moreover we show that this is the only monotone metric for which such an approach is possible

Non Symmetric Dirichlet Forms on Semifinite von Neumann Algebras

The theory of non symmetric Dirichlet forms is generalized to the non abelian setting, also establishing the natural correspondences among Dirichlet forms, sub-Markovian semigroups and sub-Markovian resolvents within this context. Examples of non symmetric Dirichlet forms given by derivations on Hilbert algebras are studied.Comment: 32 pages, plain TeX, Preprint Roma TOR VERGATA Nr.9-93-May 9

A characterisation of Wigner-Yanase skew information among statistically monotone metrics

Let M-n = M-n(C) be the space of n x n complex matrices endowed with the Hilbert-Schmidt scalar product, let S-n be the unit sphere of M-n and let D-n subset of M-n be the space of strictly positive density matrices. We show that the scalar product over D-n introduced by Gibilisco and Isola(3) (that is the scalar product induced by the map D-n There Exists rho --> rootrho is an element of S-n) coincides with the Wigner-Yanase monotone metric

Fisher information and Stam inequality on a finite group

We prove a discrete version of the Stam inequality for random variables taking values on a finite group

A dynamical uncertainty principle in von Neumann algebras by operator monotone functions

Suppose that A(1),..., A(N) are observables (selfadjoint matrices) and rho is a state (density matrix). In this case the standard uncertainty principle, proved by Robertson, gives a bound for the quantum generalized variance, namely for det{Cov(rho) (A(j), A(k) )}, using the commutators [A(j), A(k)]; this bound is trivial when N is odd. Recently a different inequality of Robertson-type has been proved by the authors with the help of the theory of operator monotone functions. In this case the bound makes use of the commutators [rho, A(j)] and is non-trivial for any N. In the present paper we generalize this new result to the von Neumann algebra case. Nevertheless the proof appears to simplify all the existing ones

New Results on Old Spectral Triples for Fractals

It is shown that many important features of nested fractals, such as the Hausdorff dimension and measure, the geodesic distance induced by the immersion in Rn (when it exists), and the self-similar energy can be recovered by the description of the fractal in terms of spectral triples. We describe in particular the case of the Vicsek square, showing that all self-similar energies can be described through a deformation of the square to a rhombus