Uncertainty inequalities in a non-Hermitian scenario beyond exact symmetry phase

Source abstract

Abstract We investigate uncertainty relations for quantum observables evolving under non-Hermitian Hamiltonians, particularly in the spontaneously broken symmetry phase and at the exceptional points (EPs). By constructing appropriate metrics in each dynamical regime, 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 EPs. 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 beyond the exact symmetry phase.