Density matrices and entropy operator for non-Hermitian quantum mechanics

TL;DR

This paper extends the concept of density matrices and entropy operators to non-Hermitian quantum mechanics, introducing Riesz Density Matrix operators and applying them to gain-loss systems and a finite-dimensional Swanson Hamiltonian.

Contribution

It introduces Riesz Density Matrix operators for non-Hermitian systems and explores their properties and applications, including systems with exceptional points.

Findings

Riesz Density Matrix operators generalize standard density matrices in non-Hermitian contexts.

Application to PT-symmetric systems demonstrates the framework's relevance.

Finite-dimensional Swanson Hamiltonian analysis reveals new insights near exceptional points.

Abstract

In this paper we consider density matrices operator related to non-Hermitian Hamiltonians. In particular, we analyse two natural extensions of what is usually called a density matrix operator (DM), of pure states and of the entropy operator: we first consider those {\em operators} which are simply similar to a standard DM, and then we discuss those which are intertwined with a DM by a third, non invertible, operator, giving rise to waht we call Riesz Density Matrix operator (RDM). After introducing the mathematical framework, we apply the framework to a couple of applications. The first application is related to a non-Hermitian Hamiltonian describing gain and loss phenomena, widely considered in the context of $PT$-quantum mechanics. The second application is related to a finite-dimensional version of the Swanson Hamiltonian, never considered before, and addresses the problem of deriving a milder version of the RDM when exceptional points form in the system.