Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts

TL;DR

This paper investigates how a family of Hermitian systems derived from a non-Hermitian quantum oscillator system exhibit universal entropy oscillations and entanglement properties, regardless of the specific Hermitian transformation applied.

Contribution

It explicitly calculates the reduced density matrix for all associated Hermitian systems and reveals universal entropy oscillation periods and entanglement behaviors independent of the Hermitian map.

Findings

Von Neumann entropy oscillates with a period twice that of the non-Hermitian system.

Oscillator and bath are entangled for almost all times.

Entanglement depends on the Hermitian map but averages out over a period.

Abstract

We consider a quantum oscillator coupled to a bath of $N$ other oscillators. The total system evolves with a quasi-Hermitian Hamiltonian. Associated to it is a family of Hermitian systems, parameterized by a unitary map $W$. Our main goal is to find the influence of $W$ on the entropy and the entanglement in the Hermitian systems. We calculate explicitly the reduced density matrix of the single oscillator for all Hermitian systems and show that, regardless of $W$, their von Neumann entropy oscillates with a common period which is twice that of the non-Hermitian system. We show that generically, the oscillator and the bath are entangled for almost all times. While the amount of entanglement depends on the choice of $W$, it is independent of $W$ when averaged over a period. These results describe some universality in the physical properties of all Hermitian systems associated to a given non-Hermitian one.