Uncertainty inequalities in a non-Hermitian scenario: the problem of the metric

• URL: http://arxiv.org/abs/2512.24437v1
• Date: Tue, 30 Dec 2025 19:42:52 GMT
• Title: Uncertainty inequalities in a non-Hermitian scenario: the problem of the metric
• Authors: Yanet Alvarez, Mariela Portesi, Romina Ramirez, Marta Reboiro,
• Abstract summary: We provide a consistent definition of expectation values, variances, and time evolution within a Krein-space framework.<n>We derive a generalized Heisenberg-Robertson uncertainty inequality which is valid across all spectral regimes.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 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 two level model with parity-time reversal symmetry and show that, while the uncertainty measure exhibits oscillatory behavior in the unbroken phase, it evolves toward a minimum-uncertainty steady state in the 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.

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