Speed limits and thermodynamic uncertainty relations for quantum systems governed by non-Hermitian Hamiltonian

TL;DR

This paper develops new speed limit and uncertainty bounds for open quantum systems governed by non-Hermitian Hamiltonians, unifying quantum speed limits with thermodynamic uncertainty relations in such systems.

Contribution

It derives novel Mandelstam-Tamm and Margolus-Levitin quantum speed limits for non-Hermitian systems, extending previous results to continuous measurement and open quantum dynamics.

Findings

Derived Mandelstam-Tamm quantum speed limit for continuous measurement.

Established Margolus-Levitin quantum speed limit in non-Hermitian systems.

Connected quantum speed limits with thermodynamic uncertainty relations.

Abstract

Non-Hermitian Hamiltonians play a crucial role in describing open quantum systems and nonequilibrium dynamics. In this paper, we derive trade-off relations for systems governed by non-Hermitian Hamiltonians, focusing on the Margolus-Levitin-type and Mandelstam-Tamm-type bounds, which are originally derived as quantum speed limits in isolated quantum dynamics. While the quantum speed limit for the Mandelstam-Tamm bound in general non-Hermitian systems was derived in the literature, we obtain a Mandelstam-Tamm quantum speed limit for continuous measurement using the continuous matrix product state formalism. Moreover, we derive a Margolus-Levitin quantum speed limit in the non-Hermitian setting. We derive additional bounds on the ratio of the standard deviation to the mean of an observable, which take the same form as the thermodynamic uncertainty relation. As an example, we apply these bounds to the continuous measurement formalism in open quantum dynamics, where the dynamics is described by discontinuous jumps and smooth evolution induced by the non-Hermitian Hamiltonian. Our work provides a unified perspective on the quantum speed limit and thermodynamic uncertainty relations in open quantum dynamics from the viewpoint of the non-Hermitian Hamiltonian, extending the results of previous studies.