Decrease of Fisher information and the information geometry of evolution equations for quantum mechanical probability amplitudes

Abstract

The relevance of the concept of Fisher information is increasing in both statistical physics and quantum computing. From a statistical mechanical standpoint, the application of Fisher information in the kinetic theory of gases is characterized by its decrease along the solutions of the Boltzmann equation for Maxwellian molecules in the two-dimensional case. From a quantum mechanical standpoint, the output state in Grover's quantum search algorithm follows a geodesic path obtained from the Fubini-Study metric on the manifold of Hilbert-space rays. Additionally, Grover's algorithm is specified by constant Fisher information. In this paper, we present an information geometric characterization of the oscillatory or monotonic behavior of statistically parametrized squared probability amplitudes originating from special functional forms of the Fisher information function: constant, exponential decay, and power-law decay. Furthermore, for each case, we compute both the computational speed and the availability loss of the corresponding physical processes by exploiting a convenient Riemannian geometrization of useful thermodynamical concepts. Finally, we briefly comment on the possibility of using the proposed methods of information geometry to help identify a suitable trade-off between speed and thermodynamic efficiency in quantum search algorithms.

Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions

“…In this paper, building upon our previous results reported in [13][14][15] and inspired by the findings uncovered in [9][10][11][12], we provide a quantitiative link between the concepts of information geometric complexity and entropic efficiency by studying the entropic dynamics on information manifolds emerging from exactly solvable time-dependent twolevel quantum systems that mimic quantum search Hamiltonians. Our motivation for considering this type of work can be explained by pointing out a number of previous results our proposed analysis relies on.…”

Information Geometric Perspective on Off-Resonance Effects in Driven Two-Level Quantum Systems

“…In Ref. [20], we presented an information geometric characterization of the oscillatory or monotonic behavior of statistically parametrized squared probability amplitudes originating from special functional forms of the Fisher information function: constant, exponential decay, and powerlaw decay. Furthermore, for each case, we computed both the computational speed and the availability loss of the corresponding physical processes by exploiting a convenient Riemannian geometrization of useful thermodynamical concepts.…”

“…Finally, building upon our works presented in Refs. [7,17,18,20], we presented in Ref. [21] an information geometric analysis of geodesic speeds and entropy production rates in geodesic motion on manifolds of parametrized quantum states.…”

“…Interestingly, being on resonance, the transition probabilities In Eqs. (20) and (22) for all four cases depend only on the integral of the transverse field intensity ω H (t). Before beginning our information geometric analysis, we formally introduce here an adimensional parameter β 0 that quantifies the departure from the on-resonance condition,…”

Information geometric complexity of entropic motion on curved statistical manifolds under different metrizations of probability spaces

“…For instance, thanks to our geodesic motion analysis together with the observed link between the information geometric complexity and the speed of convergence to the final state, our work appears to be useful for deepening our limited understanding about the existence of a tradeoff between computational speed and availability loss in an information geometric setting of quantum search algorithms with a thermodynamical flavor as presented in Refs. [35,36]. Furthermore, in view of our study of the geometrical and dynamical features that emerge from distinct metrizations of probability spaces, our comparative analysis can help investigate the unresolved problem of whether the complexity of a convex combination of two distributions is related to the complexities of the individual constituents [37].…”