Reduced dynamics in quasi-Hermitian systems

TL;DR

This paper investigates how the choice of metric operator in quasi-Hermitian systems influences the entanglement and reduced dynamics, revealing metric dependence in subsystem descriptions through a PT-symmetric quantum walk model.

Contribution

It demonstrates that the metric operator affects the entanglement structure and reduced dynamics, challenging the notion that the metric choice is physically irrelevant in non-Hermitian systems.

Findings

The partial trace of the Hermitized density matrix correctly represents the subsystem.

The metric operator influences the entanglement and non-Markovianity of the subsystem.

Subsystem decomposition depends on the chosen metric, affecting observable outcomes.

Abstract

Evolutions under non-Hermitian Hamiltonians with unbroken $\mathcal{PT}$ symmetry can be considered unitary under appropriate choices of inner products, facilitated by the so-called metric operator. While it is understood that the choice of the metric operator has no bearing on the description of the system, in this work, we show that this choice does dictate the entanglement structure of the system. We show that the partial trace of the Hermitized density matrix gives the correct representation of the reduced subsystem, and based on such operations, we elucidate the metric dependency of the reduced dynamics and consequently the observable dependence of the subsystem decomposition. We use a non-Hermitian $\mathcal{PT}$-symmetric quantum walk as a toy model to study this metric dependency, where we use the internal (coin state) as the subsystem of interest and study the coin-position entanglement and non-Markovianity of the coin dynamics.