Geometry from divergence functions and complex structures

Motivated by the geometrical structures of quantum mechanics, we introduce an almost-complex structure $J$ on the product $M\times M$ of any parallelizable statistical manifold $M$. Then, we use $J$ to extract a pre-symplectic form and a metric-like tensor on $M\times M$ from a divergence function. These tensors may be pulled back to $M$, and we compute them in the case of an N-dimensional symplex with respect to the Kullback-Leibler relative entropy, and in the case of (a suitable unfolding space of) the manifold of faithful density operators with respect to the von Neumann-Umegaki relative entropy.Comment: 19 pages, comments are welcome

Quantum States, Groups and Monotone Metric Tensors

A novel link between monotone metric tensors and actions of suitable extensions of the unitary group on the manifold of faithful quantum states is presented here by means of three illustrative examples related with the Bures-Helstrom metric tensor, the Wigner-Yanase metric tensor, and the Bogoliubov-Kubo-Mori metric tensor.Comment: 16 pages. Minor adjustments. Comments are welcome

From the Jordan product to Riemannian geometries on classical and quantum states

The Jordan product on the self-adjoint part of a finite-dimensional $C^{*}$-algebra $\mathscr{A}$ is shown to give rise to Riemannian metric tensors on suitable manifolds of states on $\mathscr{A}$, and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher--Rao metric tensor is recovered in the Abelian case, that the Fubini--Study metric tensor is recovered when we consider pure states on the algebra $\mathcal{B}(\mathcal{H})$ of linear operators on a finite-dimensional Hilbert space $\mathcal{H}$, and that the Bures--Helstrom metric tensors is recovered when we consider faithful states on $\mathcal{B}(\mathcal{H})$. Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on $\mathcal{B}(\mathcal{H})$, this alternative geometrical description clarifies the analogy between the Fubini--Study and the Bures--Helstrom metric tensor.Comment: 32 pages. Minor improvements. References added. Comments are welcome

Incompatibility in Quantum Parameter Estimation

In this paper we introduce a measure of genuine quantum incompatibility in the estimation task of multiple parameters, that has a geometric character and is backed by a clear operational interpretation. This measure is then applied to some simple systems in order to track the effect of a local depolarizing noise on the incompatibility of the estimation task. A semidefinite program is described and used to numerically compute the figure of merit when the analytical tools are not sufficient, among these we include an upper bound computable from the symmetric logarithmic derivatives only. Finally we discuss how to obtain compatible models for a general unitary encoding on a finite dimensional probe.Comment: We clarified the relation between LU and LAC measurements. 35 pages, 3 figure

Differential geometric aspects of parametric estimation theory for states on finite-dimensional C*-algebras

A geometrical formulation of estimation theory for finite-dimensional $C^{\star}$-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer-Rao and Helstrom bounds for parametric statistical models with discrete and finite outcome spaces is presented.Comment: 33 pages. Minor improvements. References added. Comments are welcome

Photonic Quantum Metrology

Quantum Metrology is one of the most promising application of quantum technologies. The aim of this research field is the estimation of unknown parameters exploiting quantum resources, whose application can lead to enhanced performances with respect to classical strategies. Several physical quantum systems can be employed to develop quantum sensors, and photonic systems represent ideal probes for a large number of metrological tasks. Here we review the basic concepts behind quantum metrology and then focus on the application of photonic technology for this task, with particular attention to phase estimation. We describe the current state of the art in the field in terms of platforms and quantum resources. Furthermore, we present the research area of multiparameter quantum metrology, where multiple parameters have to be estimated at the same time. We conclude by discussing the current experimental and theoretical challenges, and the open questions towards implementation of photonic quantum sensors with quantum-enhanced performances in the presence of noise.Comment: 51 pages, 9 figures, 967 references. Comments and feedbacks are very welcom