Non-Hermitian quantum walks and non-Markovianity: the coin-position interaction

TL;DR

This paper compares two methods of analyzing non-Hermitian, $ ext{PT}$-symmetric quantum walks, revealing differences in information flow and entanglement behavior, and highlighting the significance of pseudo-Hermiticity as a resource.

Contribution

It provides a comparative analysis of metric and normalized state formalisms for $ ext{PT}$-symmetric quantum walks, uncovering their distinct effects on information flow and entanglement.

Findings

Power law decay of information backflow indicates phase transition in metric formalism.

Differences in information flow behavior between the two methods.

Pseudo-Hermiticity can enhance entanglement, acting as a resource.

Abstract

A $\mathcal{PT}$-symmetric, non-Hermitian Hamiltonian in the $\mathcal{PT}$-unbroken regime can lead to unitary dynamics under the appropriate choice of the Hilbert space. The Hilbert space is determined by a Hamiltonian-compatible inner product map on the underlying vector space, facilitated by a ``metric operator". A more traditional method, however, involves treating the evolution as open system dynamics, and the state is constructed through normalization at each time step. In this work, we present a comparative study of the two methods of constructing the reduced dynamics of a system evolving under a $\mathcal{PT}$-symmetric Hamiltonian. Our system is a one-dimensional quantum walk with the spin and position degrees of freedom forming its two subsystems. We compare the information flow between the subsystems under the two methods. We find that under the metric formalism, a power law decay of the information backflow to the subsystem gives a clear indication of the transition from $\mathcal{PT}$-unbroken to the broken phase. This is unlike the information backflow under the normalized state method. We also note that even though non-Hermiticity models open system dynamics, pseudo-Hermiticity can increase entanglement between the subsystem in the metric Hilbert space, thus indicating that pseudo-Hermiticity cases can be seen as a resource in quantum mechanics.