Margolus-Levitin speed limit across quantum to classical regimes based on trace distance

The classical version of Mandelstam-Tamm speed limit based on the Wigner function in phase space is reported by B. Shanahan et al. [Phys. Rev. Lett. 120, 070401 (2018)]. In this paper, the Margolus-Levitin speed limit across the quantum-to-classical transition is given in phase space based on the trace distance. The Margolus-Levitin speed limit is set by the Schatten L1 norm of the generator of time dependent evolution for both the quantum and classical domains. As an example, the time-dependent harmonic oscillator is considered to illustrate the result.