Tommaso Isola

Uncertainty principle and quantum Fisher information. II.

Heisenberg and Schrodinger uncertainty principles give lower bounds for the product of variances Varρ(A)Varρ(B) if the observables A,B are not compatible, namely, if the commutator [A,B] is not zero. In this paper, we prove an uncertainty principle in Schrodinger form where the bound for the product of variances Varρ(A)Varρ(B) depends on the area spanned by the commutators i[ρ,A] and i[ρ,B] with respect to an arbitrary quantum version of the Fisher information.

Dimensions and singular traces for spectral triples, with applications to fractals

Given a spectral triple (A,H,D), the functionals on A of the form a↦τω(a|D|−α) are studied, where τω is a singular trace, and ω is a generalised limit. When τω is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|−1, and that the set of αs for which there exists a singular trace τω giving rise to a non trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The functionals corresponding to points in the traceability interval are called Hausdorff–Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff–Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit ω.

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 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 Wigner–Yanase information. Using a well-known approach that mimics the classical pull-back 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.

A trace on fractal graphs and the Ihara zeta function

Starting with Iharas work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a self-similarity property. We de. ne a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs.

A CHARACTERISATION OF WIGNER YANASE SKEW INFORMATION AMONG STATISTICALLY MONOTONE METRICS

Let be the space of n × n complex matrices endowed with the Hilbert–Schmidt 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 Isola3 (that is the scalar product induced by the map ) coincides with the Wigner–Yanase monotone metric.

CONNECTIONS ON STATISTICAL MANIFOLDS OF DENSITY OPERATORS BY GEOMETRY OF NONCOMMUTATIVE Lp-SPACES

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

Ihara's zeta function for periodic graphs and its approximation in the amenable case☆

Abstract In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi [B. Clair, S. Mokhtari-Sharghi, Zeta functions of discrete groups acting on trees, J. Algebra 237 (2001) 591–620] on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised in [R.I. Grigorchuk, A. Żuk, The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: V.A. Kaimanovich, et al. (Eds.), Proc. Workshop, Random Walks and Geometry, Vienna, 2001, de Gruyter, Berlin, 2004, pp. 141–180] by Grigorchuk and Żuk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs.

Non-symmetric Dirichlet Forms on Semifinite von Neumann Algebras

Abstract 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. Some results on the allowed functional calculus for closed derivations on Hilbert algebras are obtained. Examples of non symmetric Dirichlet forms given by derivations on Hilbert algebras are studied.

On the monotonicity of scalar curvature in classical and quantum information geometry

We study the monotonicity under mixing of the scalar curvature for the α-geometries on the simplex of probability vectors. From the results obtained and from numerical data, we are led to some conjectures about quantum α-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.

Integrals and potentials of differential 1-forms on the Sierpinski gasket

Abstract We provide a definition of the integral, along paths in the Sierpinski gasket K , for differential smooth 1-forms associated to the standard Dirichlet form E on K . We show how this tool can be used to study the potential theory on K . In particular, we prove: (i) a de Rham reconstruction of a 1-form from its periods around lacunas in K ; (ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; (iii) the existence of potentials of smooth 1-forms on a suitable covering space of K . We finally show that this framework provides versions of the de Rham duality theorem for the fractal K .