Stronger sum uncertainty relations for non-Hermitian operators

The uncertainty relations (URs) of two arbitrary Hermitian and non-Hermitian incompatible operators represented by the product of variances have been confirmed theoretically and experimentally in various physical systems. However, the lower bound of the product uncertainty inequality can be null even for two noncommuting operators, i.e., a trivial case. Therefore, for two incompatible operators over the measured system state, the associated URs regarding the sum of variances are valid in a state-dependent manner, and the lower bound is guaranteed to be nontrivial. Although the sum URs formulated for Hermitian and unitary operators have been affirmed, the general forms for arbitrary non-Hermitian operators have not yet been investigated. This study presents the sum URs for non-Hermitian operators acting on system states using an appropriate Hilbert-space metric. The compatible forms of our sum inequalities with the conventional quantum mechanics are also provided via the $G$-metric formalism. Concrete examples illustrate the validity of the proposed sum URs in both $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken phases. The developed methods and results can help give an in-depth understanding of the usefulness of $G$-metric formalism in non-Hermitian quantum mechanics and the sum URs of incompatible operators within.