Long time quantum-classical correspondence for open systems in trace norm

TL;DR

This paper demonstrates that for certain open quantum systems with quadratic Hamiltonians and linear jump functions, quantum and classical evolutions stay close in trace norm over long times, extending previous weaker norm results.

Contribution

It establishes long-time trace norm closeness between quantum and classical evolutions for specific open systems, improving upon earlier weaker norm results.

Findings

Quantum and classical evolutions remain close in trace norm over long times.

The result applies to systems with quadratic Hamiltonians and linear jump functions.

This extends previous work by strengthening the norm in which the closeness is shown.

Abstract

We consider a frictionless system coupled to an external Markovian environment. The quantum and classical evolution of such systems are described by the Lindblad and the Fokker-Planck equation respectively. We show that when such a system is given by an at most quadratically growing Hamiltonian and at most linearly growing real jump functions, the quantum and classical evolutions remain close on time scales much longer than Ehrenfest time. In particular, we show that the evolution of a density matrix by the Lindblad equation is close in trace norm to the quantization of the corresponding evolution by the Fokker-Planck equation. Such agreement improves upon recent results [arXiv:2403.09345, arXiv:2306.13717, arXiv:2307.05326], which proved long time agreement in weaker norms.