Uncertainty Relations in Non-Hermitian Systems

The Heisenberg–Robertson uncertainty relation is a standard tool in quantum mechanics, but its use in non-Hermitian settings is not straightforward, especially in regimes with complex spectra and exceptional points. In this context, we study uncertainty relations for pseudo-Hermitian, in particular PT-symmetric Hamiltonians by introducing a metric operator in all spectral regimes. As a simple and widely used test case, we focus on a non-Hermitian two-level toy model with balanced gain and loss, which shows the transition from an exact PT-symmetric phase to a symmetry-broken phase through an exceptional point.

The chosen method is based on the explicit construction, in each regime, of a positive-definite metric operator that induces a modified inner product and defines the physical expectation values and variances. On this basis, we derive generalized Heisenberg–Robertson uncertainty relations for selected observables of the two-level model and compute their behavior in the exact, broken, and exceptional-point regions. In parallel, we construct a description in terms of a Lindblad master equation for the corresponding open two-level system and use it as a reference to compare with the non-Hermitian effective dynamics.

We find that the metric-based formulation restores an uncertainty relation with the same formal structure as in the Hermitian case, whereas the direct use of the standard inner product can lead to ill-defined quantities, in particular in the symmetry-broken phase and at the exceptional point. The comparison with Lindblad dynamics shows agreement between the metric-based description and features such as PT-symmetry breaking and decoherence. These results indicate that, even for a two-level toy model, an appropriate metric is necessary to obtain consistent uncertainty bounds and dynamical predictions from pseudo-Hermitian Hamiltonians.