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Domenico Felice

Publications and source records attributed to Domenico Felice.

16 recordsLinked to original sources
arXiv2023

Geometric aspects of mixed quantum states inside the Bloch sphere

When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sj\"oqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we illustrate explicitly the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial and final mixed state when calculated with the two metrics. Interestingly, in analogy to what happens when studying topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of finite distance among mixed quantum states is not preserved when comparing distances determined with the Bures and the Sj\"oqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of complexity and volume of mixed quantum states.

quant-ph↗
arXiv2021

Towards a Canonical Divergence within Information Geometry

In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove that the rays of level-sets defined by a pseudo-distance are generated by the sum of two tangent vectors. By relying on these vectors, we propose a novel definition of a canonical divergence and its dual function. We prove that the new divergence allows to recover a given dual structure $(\mathrm{g},\nabla,\nabla^*)$ of {a dually convex set on} a smooth manifold $\mathrm{M}$. Additionally, we show that this divergence coincides with the canonical divergence proposed by Ay and Amari in the case of: (a) self-duality, (b) dual flatness, (c) statistical geometric analogue of the concept of symmetric spaces in Riemannian geometry. For a dually convex set, the case (c) leads to a further comparison of the new divergence with the one introduced by Henmi and Kobayashi.

math.DG↗
arXiv2020

Quantum Groverian Geodesic Paths with Gravitational and Thermal Analogies

We present a unifying variational calculus derivation of Groverian geodesics for both quantum state vectors and quantum probability amplitudes. In the first case, we show that horizontal affinely parametrized geodesic paths on the Hilbert space of normalized vectors emerge from the minimization of the length specified by the Fubini-Study metric on the manifold of Hilbert space rays. In the second case, we demonstrate that geodesic paths for probability amplitudes arise by minimizing the length expressed in terms of the Fisher information. In both derivations, we find that geodesic equations are described by simple harmonic oscillators (SHOs). However, while in the first derivation the frequency of oscillations is proportional to the (constant) energy dispersion of the Hamiltonian system, in the second derivation the frequency of oscillations is proportional to the square-root of the (constant) Fisher information. Interestingly, by setting these two frequencies equal to each other, we recover the well-known Anandan-Aharonov relation linking the squared speed of evolution of an Hamiltonian system with its energy dispersion. Finally, upon transitioning away from the quantum setting, we discuss the universality of the emergence of geodesic motion of SHO type in the presence of conserved quantities by analyzing two specific phenomena of gravitational and thermodynamical origin, respectively.

math-ph↗
arXiv2019

Canonical divergence for flat $α$-connections: Classical and Quantum

A recent canonical divergence, which is introduced on a smooth manifold $\mathrm{M}$ endowed with a general dualistic structure $(\mathrm{g},\nabla,\nabla^*)$, is considered for flat $α$-connections. In the classical setting, we compute such a canonical divergence on the manifold of positive measures and prove that it coincides with the classical $α$-divergence. In the quantum framework, the recent canonical divergence is evaluated for the quantum $α$-connections on the manifold of all positive definite Hermitian operators. Also in this case we obtain that the recent canonical divergence is the quantum $α$-divergence.

math-ph↗
arXiv2019

Canonical divergence for measuring classical and quantum complexity

A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both, classical and quantum systems. On the simplex of probability measures it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.

math-ph↗
arXiv2019

Divergence functions in Information Geometry

A recently introduced canonical divergence $\mathcal{D}$ for a dual structure $(\mathrm{g},\nabla,\nabla^*)$ is discussed in connection to other divergence functions. Finally, open problems concerning symmetry properties are outlined.

math.DG↗
arXiv2018

Dynamical Systems induced by Canonical Divergence in dually flat manifolds

The principles of classical mechanics have shown that the inertial quality of mass is characterized by the kinetic energy. This, in turn, establishes the connection between geometry and mechanics. We aim to exploit such a fundamental principle for information geometry entering the realm of mechanics. According to the modification of curve energy stated by Amari and Nagaoka for a smooth manifold $\mathrm{M}$ endowed with a dual structure $(\mathrm{g},\nabla,\nabla^*)$, we consider $\nabla$ and $\nabla^*$ kinetic energies. Then, we prove that a recently introduced canonical divergence and its dual function coincide with Hamilton principal functions associated with suitable Lagrangian functions when $(\mathrm{M},\mathrm{g},\nabla,\nabla^*)$ is dually flat. Corresponding dynamical systems are studied and the tangent dynamics is outlined in terms of the Riemannian gradient of the canonical divergence. Solutions of such dynamics are proved to be $\nabla$ and $\nabla^*$ geodesics connecting any two points sufficiently close to each other. Application to the standard Gaussian model is also investigated.

math-ph↗
arXiv2018

The Volume of Two-Qubit States by Information Geometry

Using the information geometry approach, we determine the volume of the set of two-qubit states with maximally disordered subsystems. Particular attention is devoted to the behavior of the volume of sub-manifolds of separable and entangled states with fixed purity. We show that the usage of the classical Fisher metric on phase space probability representation of quantum states gives the same qualitative results with respect to different versions of the quantum Fisher metric.

math-ph↗
arXiv2017

Riemannian-geometric entropy for measuring network complexity

A central issue of the science of complex systems is the quantitative characterization of complexity. In the present work we address this issue by resorting to information geometry. Actually we propose a constructive way to associate to a - in principle any - network a differentiable object (a Riemannian manifold) whose volume is used to define an entropy. The effectiveness of the latter to measure networks complexity is successfully proved through its capability of detecting a classical phase transition occurring in both random graphs and scale--free networks, as well as of characterizing small Exponential random graphs, Configuration Models and real networks.

math-ph↗
arXiv2017

Hamilton-Jacobi approach to Potential Functions in Information Geometry

The search for a potential function $S$ allowing to reconstruct a given metric tensor $g$ and a given symmetric covariant tensor $T$ on a manifold $\mathcal{M}$ is formulated as the Hamilton-Jacobi problem associated with a canonically defined Lagrangian on $T\mathcal{M}$. The connection between this problem, the geometric structure of the space of pure states of quantum mechanics, and the theory of contrast functions of classical information geometry is outlined.

math-ph↗
arXiv2017

Aspects of geodesical motion with Fisher-Rao metric: classical and quantum

The purpose of this article is to exploit the geometric structure of Quantum Mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon's Entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.

math-ph↗
arXiv2016

The volume of Gaussian states by information geometry

We formulate the problem of determining the volume of the set of Gaussian physical states in the framework of information geometry. That is, by considering phase space probability distributions parametrized by the covariances and supplying this resulting statistical manifold with the Fisher-Rao metric. We then evaluate the volume of classical, quantum and quantum entangled states for two-mode systems showing chains of strict inclusion.

math-ph↗
arXiv2016

A Study of a Nonlinear Schrödinger Equation for Optical Fibers

Non linear fiber optics concerns with the non linear optical phenomena occurring inside optical fibers. The propagation of light in single-mode fibers is governed by the one-dimensional nonlinear Schrödinger equation (NLS) in the presence of attenuation, dispersion, and non linear effects. In this NLS the role of space and time is exchanged with respect to the standard NLS as introduced in the almost entire mathematical literature. Such an exchange is far from being only formal as it enters in the interpretation of basic ideas as well-posedness and stability, as well as in the phenomenological meaning of the predictions formulated by such model. In this dissertation, the physical bases of the optical fibers are provided, and a derivation of the NLS from the Maxwell's equations is reviewed. Furthermore, problems of local nature (local existence of solutions, uniqueness) and problems of global nature (global existence) are studied by the Kato's method based on a fixed point argument and Strichartz's estimates. Moreover, tools usually employed to study global problems connected with finite-time blow up of solutions, are here used to show a closeness result between the NLS for optical fibers and an integrable NLS. Integrability is pursued by means of the Painlevé analysis that allows to describe quite a large class of integrable equations like the transformed equation from the standard NLS. Among these functions, one can be described as a standard NLS with an additional linear harmonic oscillator term. This equation is showed to be close in $L^2$-norm to the NLS for single-mode fibers. Finally, the dissertation focuses on two kinds of stationary solutions: the so-called time-homogenous, that is given by a stationary wave oscillating in space, and the soliton-like solution. Results are obtained about stability properties of them.

math-ph↗
arXiv2015

A geometric entropy detecting the Erdös-Rényi phase transition

We propose a method to associate a differentiable Riemannian manifold to a generic many degrees of freedom discrete system which is not described by a Hamiltonian function. Then, in analogy with classical Statistical Mechanics, we introduce an entropy as the logarithm of the volume of the manifold. The geometric entropy so defined is able to detect a paradigmatic phase transition occurring in random graphs theory: the appearance of the `giant component' according to the Erdös-Rényi theorem.

math-ph↗
arXiv2014

Quantifying Networks Complexity from Information Geometry Viewpoint

We consider a Gaussian statistical model whose parameter space is given by the variances of random variables. Underlying this model we identify networks by interpreting random variables as sitting on vertices and their correlations as weighted edges among vertices. We then associate to the parameter space a statistical manifold endowed with a Riemannian metric structure (that of Fisher-Rao). Going on, in analogy with the microcanonical definition of entropy in Statistical Mechanics, we introduce an entropic measure of networks complexity. We prove that it is invariant under networks isomorphism. Above all, considering networks as simplicial complexes, we evaluate this entropy on simplexes and find that it monotonically increases with their dimension.

math-ph↗