A -Adic Model of Quantum States and the -Adic Qubit.

We propose a model of a quantum N-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of -adic numbers. As in the standard complex setting, states and observables of a -adic quantum system are implemented by suitable linear operators in a -adic Hilbert space. In particular, owing to the distinguishing features of -adic probability theory, the states of an N-dimensional -adic quantum system are implemented by -adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)-a suitable -adic counterpart of a POVM in a complex Hilbert space-as a convenient mathematical tool describing the physical observables of a -adic quantum system. Eventually, we focus on the special case where N=2, thus providing a description of -adic qubit states and 2-dimensional SOVMs. The analogies-but also the non-trivial differences-with respect to the qubit states of standard quantum mechanics are then analyzed.