Quantum Speed Limits across the Quantum-to-Classical Transition

Quantum speed limits set an upper bound to the rate at which a quantum system can evolve. Adopting a phase-space approach, we explore quantum speed limits across the quantum-to-classical transition and identify equivalent bounds in the classical world. As a result, and contrary to common belief, we show that speed limits exist for both quantum and classical systems. As in the quantum domain, classical speed limits are set by a given norm of the generator of time evolution.

Figure 1

Classical speed limit to the pace of evolution. Comparison of the upper bound to the phase-space speed of evolution $v_{\mathrm{\Gamma }}^{\mathrm{CSL}}$ with the absolute value of the instantaneous rate of change of the Bhattacharyya coefficient $|\dot{B}({\varrho }_0,{\varrho }_t)|$ as a function of time. The dynamics corresponds to a free expansion of a classical probability distribution of Gaussian form that is initially confined in a harmonic potential of frequency ${\omega }_0$, which is switched off for $t>0$. The unit of time is set by ${\omega }_0^{-1}$.