Uncertainty relation for non-Hermitian systems

We construct an uncertainty relation for arbitrary finite-dimensional PT-invariant non-Hermitian quantum systems within a special inner product framework. This construction is led by "good observables," which are a more general class of operators. We show that the cumulative gain in the quantum Fisher information when measuring two good observables for such non-Hermitian systems is much better than their Hermitian counterpart. Minimum uncertainty states being the best candidates for this gain near the exceptional point supports the intelligent or simultaneous non-Hermitian quantum sensors.