Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reduced dynamics in quasi-Hermitian systems

Himanshu Badhani, C. M. Chandrashekar

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.

Reduced dynamics in quasi-Hermitian systems

Abstract

Evolutions under non-Hermitian Hamiltonians with unbroken 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 -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.
Paper Structure (12 sections, 1 theorem, 47 equations, 3 figures)

This paper contains 12 sections, 1 theorem, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Introduction to Pseudo-Hermitian quantum mechanics
  3. Hermitization of the operators
  4. Non-uniqueness of the metric and Hermitization
  5. Subsystems in $\mathcal{H}_G$
  6. Metric dependence of the subsystem:
  7. Subsystems of a quantum walk and the effect of the metric
  8. Information backflow to the coin state
  9. CP- indivisibility
  10. Conclusions
  11. Non-separability of the quantum walk metric
  12. Finding the intermediate map

Key Result

Theorem 1

Given the states $\bar{\rho}\in\mathcal{H}_G$ and $\bar{\rho}'=T^{-1}\bar{\rho}T\in\mathcal{H}_{G'}$, where $T^\dagger G T=G'$, the properties of the subsystems given by $\rho_\eta^c=\operatorname{tr}_p(\rho_\eta)$ and $\rho_{\eta'}^c=\operatorname{tr}_p(\rho_{\eta'}')$ are metric dependent unless b

Figures (3)

  • Figure 1: An unbroken $\mathcal{P}\mathcal{T}$-symmetric evolution can be seen as unitary evolution under appropriate, non-unique choices of the metric, corresponding to evolutions in different Hilbert spaces $\mathcal{H}_G$ and $\mathcal{H}_{G'}$. Operators and states in these different Hilbert spaces can be mapped in the Euclidean ($G=\mathbb{1}$) Hilbert space to unitarily equivalent states and operators.
  • Figure 2: Maximization of the BLP measure $N$ using the simulated annealing method. Each plot marks the measure evaluated after 50 steps of the walk for 3 different metric operators $G_1$, $G_2$, and $G_3$. We see that the measures saturate at roughly the same values. The Monte-Carlo method has finite precision, and within this error, the information backflow to the coin state is independent of the metric. The effect of the chosen metric is visible in another measure of non-Markovianity.
  • Figure 3: \ref{['rhpfig']} RHP measure as a function of time for the reduced dynamics under different choices of metric operators. We see clearly that the CP-indivisibility of the coin state's dynamical map in the Euclidean metric space has different measure values for different choices of the metric in the $\delta\neq 1$ case. This is a reflection of the non-separability of the metric operator in the non-trivial metric space. For the Hermitian case, we do not see any metric dependency. \ref{['entfig']} Coin-position entanglement as a function of time under different choices of the metric operator. The coin-position entanglement in the Euclidean metric space shows the metric dependency for dynamics with $\delta\neq 1$.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • proof