Information Geometric Aspects of Probability Paths with Minimum Entropy Production for Quantum State Evolution

TL;DR

This paper uses information geometry to analyze quantum state evolution, showing that optimal, shortest paths minimize entropy production and that faster transfers increase entropy rate, revealing a trade-off in quantum information processes.

Contribution

It introduces an information geometric framework for analyzing quantum state transfer paths, linking minimal entropy production with geodesic paths in parameter space.

Findings

Optimal paths are geodesics minimizing entropy production.

Faster quantum state transfer correlates with higher entropy production rate.

Higher entropic speed implies lower entropic efficiency in quantum transfers.

Abstract

We present an information geometric analysis of both entropic speeds and entropy production rates arising from geodesic evolution on manifolds parametrized by pure quantum states. In particular, we employ pure states that emerge as outputs of suitably chosen su(2; C) time-dependent Hamiltonian operators that characterize analog quantum search algorithms of specific types. The su(2; C) Hamiltonian models under consideration are specified by external time-dependent magnetic fields within which spin-1/2 test particles are immersed. The positive definite Riemannian metrization of the parameter manifold is furnished by the Fisher information function. The Fisher information function is evaluated along parametrized squared probability amplitudes obtained from the temporal evolution of these spin-1/2 test particles. A minimum action approach is then utilized to induce the transfer of the quantum system from its initial state to its final state on the parameter manifold over a finite temporal interval. We demonstrate in an explicit manner that the minimal (that is, optimum) path corresponds to the shortest (that is, geodesic) path between the initial and final states. Furthermore, we show that the minimal path serves also to minimize the total entropy production occurring during the transfer of states. Finally, upon evaluating the entropic speed as well as the total entropy production along optimal transfer paths within several scenarios of physical interest in analog quantum searching algorithms, we demonstrate in a transparent quantitative manner a correspondence between a faster transfer and a higher rate of entropy production. We therefore conclude that higher entropic speed is associated with lower entropic efficiency within the context of quantum state transfer.