Composing $p$-adic qubits: from representations of SO(3)$_p$ to entanglement and universal quantum logic gates

• URL: http://arxiv.org/abs/2601.13808v1
• Date: Tue, 20 Jan 2026 10:07:49 GMT
• Title: Composing $p$-adic qubits: from representations of SO(3)$_p$ to entanglement and universal quantum logic gates
• Authors: Ilaria Svampa, Sonia L'Innocente, Stefano Mancini, Andreas Winter,
• Abstract summary: We build on the notion of a single $p$-adic qubit as a two-dimensional irreducible representation of the compact $p$-adic special group SO(3)$_p$.<n>We tackle the problem of $p$-adic qubit composition and entanglement, fundamental for a $p$-adic formulation of quantum information processing.
• Score: 1.3999481573773072
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In the context of $p$-adic quantum mechanics, we investigate composite systems of $p$-adic qubits and $p$-adically controlled quantum logic gates. We build on the notion of a single $p$-adic qubit as a two-dimensional irreducible representation of the compact $p$-adic special orthogonal group SO(3)$_p$. We show that the classification of these representations reduces to the finite case, as they all factorise through some finite quotient SO(3)$_p$ mod $p^k$. Then, we tackle the problem of $p$-adic qubit composition and entanglement, fundamental for a $p$-adic formulation of quantum information processing. We classify the representations of SO(3)$_p$ mod $p$, and analyse tensor products of two $p$-adic qubit representations lifted from SO(3)$_p$ mod $p$. We solve the Clebsch-Gordan problem for such systems, revealing that the coupled bases decompose into singlet and doublet states. We further study entanglement arising from those stable subsystems. For $p=3$, we construct a set of gates from $4$-dimensional irreducible representations of SO(3)$_p$ mod $p$ that we prove to be universal for quantum computation.

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