An approach to $p$-adic qubits from irreducible representations of $SO(3)_p$

Abstract

We introduce the notion of $p$-adic quantum bit ($p$-qubit) in the context of the $p$-adic quantum mechanics initiated and developed by Volovich and his followers. In this approach, physics takes place in three-dimensional $p$-adic space rather than Euclidean space. Based on our prior work describing the $p$-adic special orthogonal group, we outline a programme to classify its continuous unitary projective representations, which can be interpreted as a theory of $p$-adic angular momentum. The $p$-adic quantum bit arises from the irreducible representations of minimal nontrivial dimension two, of which we construct examples for all primes $p$.