Unifying Quantum and Classical Speed Limits on Observables

TL;DR

This paper derives a unified bound on the speed of observable evolution in open quantum systems, generalizing classical and quantum speed limits, and introduces a framework to characterize and optimize these limits.

Contribution

It introduces a unified speed limit for open quantum systems, combining classical and quantum uncertainty principles, and provides methods to identify observables that saturate these limits.

Findings

Tighter bounds on observable evolution speed than previous quantum speed limits.

A basis of speed operators characterizes observables that reach the speed limit.

Bounds on incoherent dynamics' effect and optimal Hamiltonians for maximum speedup.

Abstract

The presence of noise or the interaction with an environment can radically change the dynamics of observables of an otherwise isolated quantum system. We derive a bound on the speed with which observables of open quantum systems evolve. This speed limit divides into Mandalestam and Tamm's original time-energy uncertainty relation and a time-information uncertainty relation recently derived for classical systems, generalizing both to open quantum systems. By isolating the coherent and incoherent contributions to the system dynamics, we derive both lower and upper bounds to the speed of evolution. We prove that the latter provide tighter limits on the speed of observables than previously known quantum speed limits, and that a preferred basis of \emph{speed operators} serves to completely characterize the observables that saturate the speed limits. We use this construction to bound the effect of incoherent dynamics on the evolution of an observable and to find the Hamiltonian that gives the maximum coherent speedup to the evolution of an observable.