Quantum Groverian Geodesic Paths with Gravitational and Thermal Analogies

TL;DR

This paper unifies the derivation of quantum geodesic paths using variational calculus, revealing simple harmonic oscillator equations and connecting quantum evolution speed with energy and Fisher information, with implications for gravitational and thermodynamical systems.

Contribution

It provides a unified variational derivation of quantum geodesics for states and probabilities, linking them through SHO equations and extending the concept to gravitational and thermodynamical analogies.

Findings

Geodesic equations are described by simple harmonic oscillators.

Frequency of oscillations relates to energy dispersion and Fisher information.

Universal emergence of SHO-type geodesics in systems with conserved quantities.

Abstract

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.