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

TL;DR

This paper introduces $p$-adic qubits within a $p$-adic quantum mechanics framework, utilizing irreducible representations of $SO(3)_p$ to define quantum bits in a non-Archimedean setting.

Contribution

It develops a classification of continuous unitary projective representations of $SO(3)_p$, leading to the construction of $p$-adic qubits for all primes $p$.

Findings

Constructed examples of $p$-adic qubits for all primes $p$.

Outlined a classification scheme for $p$-adic angular momentum representations.

Proposed a new $p$-adic quantum information framework.

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 $p$.