Coherent states of the p-adic Heisenberg group, heterodyne measurements and entropy uncertainty relation

A construction of coherent states for the p-adic Heisenberg group is proposed. It turns out that these states are parametrized by elements of the direct product of two copies of the Prufer group and form an orthonormal basis. Such bases are parametrized by a set of self-dual lattices in a two-dimensional vector space over a field of p-adic numbers. The whole family of coherent state bases can be represented as a graph of self-dual lattices. The heterodyne measurement corresponding to this basis is investigated. For a pair of such measurements, the uncertainty relation is obtained in terms of the Wehrl entropy. The lower bound of the sum of the Wehrl entropies for two dimensions is determined by the distance on the lattice graph. It is shown that the bases of coherent states are pairwise mutually unbiased. A complete system of mutually unbiased bases is constructed.