Uncertainty inequalities in a non-Hermitian scenario

TL;DR

This paper develops a framework for uncertainty relations in non-Hermitian quantum systems using metric operators, providing a generalized Heisenberg-Robertson inequality valid across different spectral regimes.

Contribution

It introduces a consistent metric-based approach to define expectation values and variances in non-Hermitian quantum dynamics, extending uncertainty relations to all spectral phases.

Findings

Uncertainty measure oscillates in unbroken phase

Steady state minimizes uncertainty in broken and exceptional phases

Metric-based approach aligns with Lindblad dynamics in steady state

Abstract

We investigate uncertainty relations for quantum observables evolving under non-Hermitian Hamiltonians, with particular emphasis on the role of metric operators. By constructing appropriate metrics in each dynamical regime, namely the unbroken-symmetry phase, the spontaneously broken-symmetry phase, and at exceptional points, we provide a consistent definition of expectation values, variances, and time evolution within a Krein-space framework. Within this approach, we derive a generalized Heisenberg-Robertson uncertainty inequality which is valid across all spectral regimes. As an application, we analyze a spin model with parity-time reversal symmetry and show that, while the uncertainty measure exhibits oscillatory behavior in the unbroken phase, it evolves towards a minimum-uncertainty steady state in the spontaneously broken-symmetry phase and at exceptional points. We further compare our metric-based description with a Lindblad master-equation approach and show their agreement in the steady state. Our results highlight the necessity of incorporating appropriate metric structures to extract physically meaningful predictions from non-Hermitian quantum dynamics.