1 research outputs found
Decrease of Fisher information and the information geometry of evolution equations for quantum mechanical probability amplitudes
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.Comment: 42 pages, 3 figures, 1 tabl