Mixedness timescale in non-Hermitian quantum systems

TL;DR

This paper derives timescales for the growth of mixedness in non-Hermitian quantum systems, providing insights into their short-time dynamics and applications in quantum sensing and thermodynamics.

Contribution

It introduces a perturbative approach to quantify mixedness timescales in non-Hermitian quantum systems, including bipartite and many-body models, with minimal computational effort.

Findings

Non-Hermitian Hamiltonians accelerate mixedness growth.

Derived timescales recover Hermitian coherence loss in the Hermitian limit.

Application to many-body models shows enhanced short-time entropy dynamics.

Abstract

We discuss the short-time perturbative expansion of the linear entropy for finite-dimensional quantum systems whose dynamics can be effectively described by a non-Hermitian Hamiltonian. We derive a timescale for the degree of mixedness for an input state undergoing non-Hermitian dynamics and specialize these results in the case of a driven-dissipative two-level system. Next, we derive a timescale for the growth of mixedness for bipartite quantum systems that depends on the effective non-Hermitian Hamiltonian. In the Hermitian limit, this result recovers the perturbative expansion for coherence loss in Hermitian systems, while it provides an entanglement timescale for initial pure and uncorrelated states. To illustrate these findings, we consider the many-body transverse-field $XY$ Hamiltonian coupled to an imaginary all-to-all Ising model. We find that the non-Hermitian Hamiltonian enhances the short-time dynamics of the linear entropy for the considered input states. Overall, each timescale depends on minimal ingredients such as the probe state and the non-Hermitian Hamiltonian of the system, and its evaluation requires low computational cost. Our results find applications to non-Hermitian quantum sensing, quantum thermodynamics of non-Hermitian systems, and $\mathcal{PT}$-symmetric quantum field theory.