arXiv subjects
Publications and source records attributed to Franco Ventriglia.
We present systematic proofs of statements about probability representations of qudit density states in terms of standard probability distributions of dichotomic random variables. New relations and new entropic-information inequalities are derived. The examples of 3- and 4- level states are explicitly worked out.
Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group $ϕ_t$ of transformations on a Hilbert space, $\mathcal{H}$, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schrödinger equation to a submanifold invariant under the flow $ϕ_t$. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow $ϕ_t$. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.
We give a Lagrangian description of an electric charge in a field sourced by a continuous magnetic monopole distribution. The description is made possible thanks to a doubling of the configuration space. The Legendre transform of the nonrelativistic Lagrangian agrees with the Hamiltonian description given recently by Kupriyanov and Szabo. The covariant relativistic version of the Lagrangian is shown to introduce a new gauge symmetry, in addition to standard reparametrizations. The generalization of the system to open strings coupled to a magnetic monopole distribution is also given, as well as the generalization to particles in a non-Abelian gauge field which does not satisfy Bianchi identities in some region of the space-time.
We review the geometry of the space of quantum states $\mathscr{S}(\mathcal{H})$ of a finite-level quantum system with Hilbert space $\mathcal{H}$ from a group-theoretical point of view. This space carries two stratifications generated by the action of two different Lie groups: the special unitary group $\mathcal{SU}(\mathcal{H})$ and its complexification $\mathcal{SL}(\mathcal{H})$, the complex special linear group. A stratum of the stratification generated by $\mathcal{SU}(\mathcal{H})$ is composed of isospectral states, that is, density operators with the same spectrum, A stratum of the stratification generated by $\mathcal{SL}(\mathcal{H})$ is composed of quantum states with the same rank. We prove that on every submanifold of isospectral quantum states there is also a canonical left action of $\mathcal{SL}(\mathcal{H})$ which is related with the canonical Kähler structure on isospectral quantum states. The fundamental vector fields of this $\mathcal{SL}(\mathcal{H})$-action are divided into Hamiltonian and gradient vector fields. The former give rise to invertible maps on $\mathscr{S}(\mathcal{H})$ that preserve the von Neumann entropy and the convex structure of $\mathscr{S}(\mathcal{H})$, while the latter give rise to invertible maps on $\mathscr{S}(\mathcal{H})$ that preserve the von Neumann entropy but not the convex structure of $\mathscr{S}(\mathcal{H})$. A similar decomposition is given for the $\mathcal{SL}(\mathcal{H})$-action generating the stratification of $\mathscr{S}(\mathcal{H})$ into manifolds of quantum states with the same rank, where gradient vector fields preserve the rank but do not preserve entropy. Some comments on multipartite quantum systems are made. It is proved that the sets of product states of a multipartite quantum system are homogeneous manifolds for the action of the complex special linear group associated with the partition.
The so-called $q$-z-\textit{Rényi Relative Entropies} provide a huge two-parameter family of relative entropies which includes almost all well-known examples of quantum relative entropies for suitable values of the parameters. In this paper we consider a log-regularized version of this family and use it as a family of potential functions to generate covariant $(0,2)$ symmetric tensors on the space of invertible quantum states in finite dimensions. The geometric formalism developed here allows us to obtain the explicit expressions of such tensor fields in terms of a basis of globally defined differential forms on a suitable unfolding space without the need to introduce a specific set of coordinates. To make the reader acquainted with the intrinsic formalism introduced, we first perform the computation for the qubit case, and then, we extend the computation of the metric-like tensors to a generic $n$-level system. By suitably varying the parameters $q$ and $z$, we are able to recover well-known examples of quantum metric tensors that, in our treatment, appear written in terms of globally defined geometrical objects that do not depend on the coordinates system used. In particular, we obtain a coordinate-free expression for the von Neumann-Umegaki metric, for the Bures metric and for the Wigner-Yanase metric in the arbitrary $n$-level case.
In the framework of quantum information geometry we investigate the relationship between monotone metric tensors uniquely defined on the space of quantum tomograms, once the tomographic scheme chosen, and monotone quantum metrics on the space of quantum states, classified by operator monotone functions, according to Petz classification theorem. We show that different metrics can be related through a change of the tomographic map and prove that there exists a bijective relation between monotone quantum metrics associated with different operator monotone functions. Such bijective relation is uniquely defined in terms of solutions of a first order second degree differential equation for the parameters of the involved tomographic maps. We first exhibit an example of a non-linear tomographic map which connects a monotone metric with a new one which is not monotone. Then we provide a second example where two monotone metrics are uniquely related through their tomographic parameters.
In the framework of quantum information geometry, we derive, from quantum relative Tsallis entropy, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential calculus. The cases N = 2, N = 3 are discussed in detail and notable limits are analyzed. The radial limit procedure has been used to recover quantum metrics for lower rank states, such as pure states. By using the tomographic picture of quantum mechanics we have obtained the Fisher- Rao metric for the space of quantum tomograms and derived a reconstruction formula of the quantum metric of density states out of the tomographic one. A new inequality obtained for probabilities of three spin-1/2 projections in three perpendicular directions is proposed to be checked in experiments with superconducting circuits.
Based on a geometric picture, the example of free particle motion for both classical and quantum domains is considered in the tomographic probability representation. Wave functions and density operators as well as optical and symplectic tomograms are obtained as solutions of kinetic classical and quantum equations for the state tomograms. The difference of tomograms of free particle for classical and quantum states is discussed.
The tomographic picture of quantum mechanics has brought the description of quantum states closer to that of classical probability and statistics. On the other hand, the geometrical formulation of quantum mechanics introduces a metric tensor and a symplectic tensor (Hermitian tensor) on the space of pure states. By putting these two aspects together, we show that the Fisher information metric, both classical and quantum, can be described by means of the Hermitian tensor on the manifold of pure states.