Uncertainty in Non-Hermitian Quantum Systems

This paper addresses the fundamental problem of defining and understanding uncertainty relations in quantum systems described by non-Hermitian Hamiltonians. This is crucial because non-Hermitian Hamiltonians are used to model open quantum systems and systems with gain and loss, which are increasingly important in areas like quantum optics and condensed matter physics. The paper's focus on the role of metric operators and its derivation of a generalized Heisenberg-Robertson uncertainty inequality across different spectral regimes is a significant contribution. The comparison with the Lindblad master-equation approach further strengthens the paper's impact by providing a link to established methods.

• •Provides a consistent definition of expectation values, variances, and time evolution within a Krein-space framework for non-Hermitian quantum systems.
• •Derives a generalized Heisenberg-Robertson uncertainty inequality applicable across all spectral regimes.
• •Demonstrates the importance of incorporating appropriate metric structures for physically meaningful predictions in non-Hermitian quantum dynamics.
• •Compares the metric-based description with a Lindblad master-equation approach, showing agreement in the steady state.