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Density matrices and entropy operator for non-Hermitian quantum mechanics

Fabio Bagarello, Francesco Gargano, Lidia Saluto

TL;DR

This work develops a formal framework to extend density matrices, entropy, and pure-state notions to non-Hermitian quantum mechanics by introducing two constructions: (i) Riesz density matrices (RDMs) $ρ=Rρ_0R^{-1}$ generated by a bounded invertible operator $R$, and (ii) generalized density matrices (GDMs) defined by an intertwining relation $ρR=Rρ_0$ with non-invertible $R$. The authors define corresponding pure-state and entropy concepts (RPS, GPS, GEO) and analyze their mathematical properties, including trace, positivity, and bi-orthogonal bases. They validate the framework on two finite-dimensional models: a two-state gain/loss system and a three-level Swanson-inspired model, examining how trace, purity, and entropy behave near exceptional points (EPs). The results show that RDMs preserve trace, purity, and entropy under similarity, while GDMs can lose unit trace and require normalization, with EPs driving significant changes in the mixedness and entropy of the states, suggesting new avenues for non-Hermitian state analysis in both finite and potentially infinite dimensions.

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.

Density matrices and entropy operator for non-Hermitian quantum mechanics

TL;DR

This work develops a formal framework to extend density matrices, entropy, and pure-state notions to non-Hermitian quantum mechanics by introducing two constructions: (i) Riesz density matrices (RDMs) generated by a bounded invertible operator , and (ii) generalized density matrices (GDMs) defined by an intertwining relation with non-invertible . The authors define corresponding pure-state and entropy concepts (RPS, GPS, GEO) and analyze their mathematical properties, including trace, positivity, and bi-orthogonal bases. They validate the framework on two finite-dimensional models: a two-state gain/loss system and a three-level Swanson-inspired model, examining how trace, purity, and entropy behave near exceptional points (EPs). The results show that RDMs preserve trace, purity, and entropy under similarity, while GDMs can lose unit trace and require normalization, with EPs driving significant changes in the mixedness and entropy of the states, suggesting new avenues for non-Hermitian state analysis in both finite and potentially infinite dimensions.

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 -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.
Paper Structure (14 sections, 5 theorems, 62 equations, 3 figures)

This paper contains 14 sections, 5 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Density matrices and pure states
  3. A short review for $\rho_0=\rho_0^\dagger$
  4. Similarity operators and DMs
  5. Intertwining operators and DMs
  6. Examples of generalized DMs
  7. Application I: a two-state non-Hermitian system
  8. Density matrices
  9. Application II: The finite dimensional Swanson model
  10. RDM I: a time dependent case
  11. Entropy and purity
  12. RDM II: a time independent case
  13. GDM in time independent case
  14. Conclusions

Key Result

Theorem 2

A DM $\rho$ is a pure state if and only if one of the following properties, all equivalent, is satisfied: p1.$tr(\rho^2)=1$. p2.$S(\rho)=0$. p3.$\rho$ is an extremal point of ${\cal G}$.

Figures (3)

  • Figure 1: (a) Behavior of purity $tr(\rho^2(t))$ and entropy $tr(S(\rho(t)))$ in the vicinity of the exceptional point $\alpha_2 = -1/2$, with $\alpha_1 = 1$ for the RDM I example. At the exceptional point, $R$ is not invertible, and the purity tends to its minimal value of 1/3, while the entropy to its maximal value of $\log(3)$. (b) Behavior of purity and entropy for decreasing values of $\alpha_2$ with $\alpha_1 = 1$. As $\alpha_2$ moves away from the exceptional point, the purity and entropy approach their asymptotic values of $1/2$ and $\log(2)$, respectively. (c) Same as (b) but for increasing $\alpha_2$.
  • Figure 2: (a) Behavior of the purity $tr(\rho^2)$ for various value of $\beta$ and with $\alpha>-1/2$, $\alpha_1=1$ for the RDM II example. As $\alpha_2\rightarrow -1/2$, $\rho$ tends to a Riesz pure state, whereas for $\alpha_2\rightarrow \infty$, $\rho$ is a fully mixed state. (b)) Same as (a) but for the entropy $tr(S(\rho))$.
  • Figure 3: Behavior of purity $tr(\rho^2)$ and entropy $tr(S(\rho))$ as function of the parameter $\lambda_1$ for the GDM example .

Theorems & Definitions (11)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Definition 5
  • Theorem 6
  • Definition 7
  • Proposition 8
  • Definition 9
  • Definition 10
  • ...and 1 more