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 analogs of Fisher information. Although there exists a number of general theorems shedding 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 WignerYanase information. Using a well-known approach that mimics the classical pullback approach to Fisher information, we are able to give explicit formulas 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.

The non-parametric version of Information Geometry has been developed in recent years. The first basic result was the construction of the manifold structure on ℳμ, the maximal statistical models associated to an arbitrary measure μ (see Ref. 48). Using this construction we first show in this paper that the pretangent and the tangent bundles on ℳμ are the natural domains for the mixture connection and for its dual, the exponential connection. Second we show how to define a generalized Amari embedding AΦ:ℳμ→SΦ from the Exponential Statistical Manifold (ESM) ℳμ to the unit sphere SΦ of an arbitrary Orlicz space LΦ. Finally we show that, in the non-parametric case, the α-connections ∇α(α∈(-1,1)) must be defined on a suitable α-bundle ℱα over ℳμ and that the bundle-connection pair (ℱα, ∇α) is simply (isomorphic to) the pull-back of the Amari embedding Aα: ℳμ→S2/1-α were the unit sphere S2/1-αcL2/1-α is equipped with the natural connection.

In this paper the relation between quantum covariances and quantum Fisher informations are studied. This study is applied to generalize a recently proved uncertainty relation based on quantum Fisher information. The proof given here considerably simplify the previously proposed proofs and leads to more general inequalities.2000 Mathematics Subject Classi cation. Primary 62B10, 94A17; Secondary 46L30, 46L60.

Heisenberg and Schrödinger uncertainty principles give lower bounds for the product of variances 2000 Mathematics Subject Classification. Primary 62B10, 94A17; Secondary 46L30, 46L60.

Heisenberg and Schrödinger uncertainty principles give lower bounds for the product of variances Var ͑A͒Var ͑B͒ if the observables A , B are not compatible, namely, if the commutator ͓A , B͔ is not zero. In this paper, we prove an uncertainty principle in Schrödinger form where the bound for the product of variances Var ͑A͒Var ͑B͒ depends on the area spanned by the commutators i͓ , A͔ and i͓ , B͔ with respect to an arbitrary quantum version of the Fisher information.

Let Mn = Mn(C) be the space of n × n complex matrices endowed with the HilbertSchmidt scalar product, let Sn be the unit sphere of Mn and let Dn ⊂ Mn be the space of strictly positive density matrices. We show that the scalar product over Dn introduced by Gibilisco and Isola 3 (that is the scalar product induced by the map Dn ρ → √ ρ ∈ Sn)coincides with the Wigner-Yanase monotone metric.

Let N be a statistical manifold of density operators, with respect to an n.s.f. trace τ on a semifinite von Neumann algebra M . If S p is the unit sphere of the noncommutative space L p (M, τ ), using the noncommutative Amari embedding ρ ∈ N → ρ 1/p ∈ S p , we define a noncommutative α-bundle-connection pair (F α , ∇ α ), by the pullback technique. In the commutative case we show that it coincides with the construction of nonparametric Amari-Čentsov α-connection made in Ref. 8 by Gibilisco and Pistone.

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