From Uncertainty Relations to Quantum Acceleration Limits

TL;DR

This paper compares different uncertainty relation-based methods to derive quantum acceleration limits, analyzes conditions for maximum acceleration in two-level systems, and discusses extensions to higher-dimensional quantum systems.

Contribution

It provides a comparative analysis of Robertson and Robertson-Schrödinger uncertainty relations in deriving quantum acceleration bounds and explicitly characterizes conditions for maximal acceleration in two-level systems.

Findings

Derived explicit conditions for maximal quantum acceleration on the Bloch sphere.

Compared the Robertson and Robertson-Schrödinger uncertainty relation approaches.

Illustrated findings with physical examples involving time-varying magnetic fields.

Abstract

The concept of quantum acceleration limit has been recently introduced for any unitary time evolution of quantum systems under arbitrary nonstationary Hamiltonians. While Alsing and Cafaro [Int. J. Geom. Methods Mod. Phys. 21, 2440009 (2024)] used the Robertson uncertainty relation in their derivation, Pati [arXiv:quant-ph/2312.00864 (2023)] employed the Robertson-Schr\"odinger uncertainty relation to find the upper bound on the temporal rate of change of the speed of quantum evolutions. In this paper, we provide a comparative analysis of these two alternative derivations for quantum systems specified by an arbitrary finite-dimensional projective Hilbert space. Furthermore, focusing on a geometric description of the quantum evolution of two-level quantum systems on a Bloch sphere under general time-dependent Hamiltonians, we find the most general conditions needed to attain the maximal upper bounds on the acceleration of the quantum evolution. In particular, these conditions are expressed explicitly in terms of two three-dimensional real vectors, the Bloch vector that corresponds to the evolving quantum state and the magnetic field vector that specifies the Hermitian Hamiltonian of the system. For pedagogical reasons, we illustrate our general findings for two-level quantum systems in explicit physical examples characterized by specific time-varying magnetic field configurations. Finally, we briefly comment on the extension of our considerations to higher-dimensional physical systems in both pure and mixed quantum states.