Entropic uncertainty relations and quasi-Hermitian operators

We discuss a possible treatment of quasi-Hermitian operators from the viewpoint of the uncertainty principle. Here, probabilities are actually determined by the pair containing the square root of a given metric operator and adopted resolution of the identity. For two pairs of such a kind, we derive some inequality between norm-like functionals of generated probability distributions. Based on Riesz's theorem, this inequality assumes that one enjoys some condition with norms for the squared roots of metric operators and measured density matrix. The derived inequality between norm-like functionals naturally leads to entropic uncertainty relations in terms of the unified entropies. Entropic bounds of both the state-dependent and state-independent forms are presented. The latter form means some implicit dependence, since the measured density matrix is involved in the above condition. The presented entropic bounds are an extension of the previous bounds to the quasi-Hermitian case. The results are discussed within an example of 2×2 quasi-Hermitian matrices. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'. (paper)