Speed Limits for Macroscopic Transitions

TL;DR

This paper introduces a new framework for deriving tighter speed limits on macroscopic state transitions in quantum systems, based on local conservation laws, with implications for nonequilibrium dynamics and quantum control.

Contribution

It develops a general method to establish tighter speed bounds for macroscopic observables using local conservation laws, surpassing traditional limits like Lieb-Robinson bounds.

Findings

Derived a speed limit based on the observable's gradient.

Showed the speed limit decreases with the transition Hamiltonian's expectation value.

Achieved bounds that depend on instantaneous quantum states, allowing equality conditions.

Abstract

Speed of state transitions in macroscopic systems is a crucial concept for foundations of nonequilibrium statistical mechanics as well as various applications in quantum technology represented by optimal quantum control. While extensive studies have made efforts to obtain rigorous constraints on dynamical processes since Mandelstam and Tamm, speed limits that provide tight bounds for macroscopic transitions have remained elusive. Here, by employing the local conservation law of probability, the fundamental principle in physics, we develop a general framework for deriving qualitatively tighter speed limits for macroscopic systems than many conventional ones. We show for the first time that the speed of the expectation value of an observable defined on an arbitrary graph, which can describe general many-body systems, is bounded by the "gradient" of the observable, in contrast with conventional speed limits depending on the entire range of the observable. This framework enables us to derive novel quantum speed limits for macroscopic unitary dynamics. Unlike previous bounds, the speed limit decreases when the expectation value of the transition Hamiltonian increases; this intuitively describes a new tradeoff relation between time and quantum phase difference. Our bound is dependent on instantaneous quantum states and thus can achieve the equality condition, which is conceptually distinct from the Lieb-Robinson bound. Our work elucidates novel speed limits on the basis of local conservation law, providing fundamental limits to various types of nonequilibrium quantum macroscopic phenomena.