p-adic Qubit Composition

• p-adic qubit composition is a framework that models quantum bits using non-Archimedean fields, ultrametric norms, and tensor products of p-adic Hilbert spaces.
• Its representation theory leverages compact p-adic Lie groups and Clebsch–Gordan decompositions to construct composite systems and simulate Bell-type entanglement.
• The approach enables novel quantum gate operations and protocols with unique error correction and cryptographic potential in quantum information theory.

A $p$-adic qubit is a mathematical abstraction of quantum information in the setting of non-Archimedean local fields, most commonly $\Q_p$ or a finite extension, equipped with ultrametric norms and inner products. The formalism of $p$-adic qubit composition draws upon the representation theory of compact $p$-adic Lie groups, tensor products of $p$-adic Hilbert spaces, and ultrametric functional analysis. This framework enables one to define multi-qubit systems, gates, entanglement, and universal logic operations, providing a foundation for $p$-adic quantum information theory distinct from its complex counterpart.

1. $p$-adic Qubit Fundamentals

A single $p$-adic qubit is realized as a two-dimensional irreducible representation (irrep) of the compact $p$-adic rotation group $SO(3)_p$. Explicitly, $SO(3)_p$ is the subgroup of $3\times3$ matrices over $\Q_p$ preserving a non-degenerate quadratic form $A_+$ (depending on $p$) and with determinant one. The representation theory of $SO(3)_p$ is profinite: every finite-dimensional irrep factors through a quotient $SO(3)_p \bmod p^k$ for some $k\geq1$. All two-dimensional irreps arise from $G_p=SO(3)_p \bmod p$ and are indexed by $j=1,\dots,(p-1)/2$ when $p>2$ (Svampa et al., 20 Jan 2026).

In standard vector space language, the state space of a $p$-adic qubit is $H_p=Q_p|0\rangle + Q_p|1\rangle$ with orthonormal computational basis vectors $|0\rangle=(1,0)^T$, $|1\rangle=(0,1)^T$ (Weng, 2021). The $p$-adic inner product is given as a $\Q_p$-valued dot product. The norm is the non-Archimedean sup-norm on coordinates: $\|\psi\|_p = \max_x |\psi_x|_p$.

2. Tensor Products of $p$-adic Hilbert Spaces

Composite $p$-adic systems are constructed via tensor products of $p$-adic Hilbert spaces. Let $H_1,H_2$ be vector spaces over a quadratic extension $Q/\Q_p$, each with ultrametric norm and $p$-adic inner product (Aniello et al., 8 Oct 2025). The algebraic tensor product $H_1 \otimes_{\mathrm{alg}} H_2$ is defined by the standard universal property: every bilinear map $H_1\times H_2 \to Z$ factors through the tensor product. The natural norm on this product is the $p$-adic projective norm, a seminorm defined for $u=\sum_{i=1}^{n} x_i\otimes y_i$ as

$$\|u\|_\pi = \inf\left\{ \max_{1\leq i\leq n}\|x_i\|\|y_i\|\,:\,u=\sum_{i=1}^n x_i\otimes y_i \right\}$$

reflecting the ultrametric triangle law; for a simple tensor, $\|x\otimes y\|_\pi = \|x\|\|y\|$.

The completion of the quotient by the zero-norm subspace yields a $p$-adic Banach space denoted $H_1\widehat{\otimes}_\pi H_2$. On simple tensors, the inner product is defined by $$\langle x_1\otimes y_1, x_2\otimes y_2\rangle_\otimes = \langle x_1,x_2\rangle_{H_1}\,\langle y_1, y_2\rangle_{H_2}$$, extended by bilinearity and continuity. The orthonormal basis of $H_1\widehat{\otimes}_\pi H_2$ is given by $\{\phi_i\otimes \psi_j\}$ if $\{\phi_i\}$ and $\{\psi_j\}$ are ON bases of $H_1$ and $H_2$.

3. Representation-Theoretic Composition and Clebsch–Gordan Decomposition

The composition of $p$-adic qubits is governed by the tensor product of irreps. For qubit irreps $\rho_A,\rho_B:G_p\to O(2)$, the composite system is the tensor-product representation $\rho_A\otimes \rho_B:G_p\to U(4)$. The Clebsch–Gordan problem over $G_p$ (a semidirect product $C_p^2\rtimes D_{p+1}$) reduces decomposition to its $D_{p+1}$ subquotient:

• If $j=\ell$, then $$\sigma^{(j)}\otimes\sigma^{(j)} \cong \chi^{\rm triv} \oplus \chi^{(1)} \oplus \sigma^{(2j)}$$.
• If $j+\ell\equiv p+1\pmod{2(p+1)}$, then $$\sigma^{(j)}\otimes\sigma^{(p+1-j)} \cong \chi^{(2)} \oplus \chi^{(3)} \oplus \sigma^{(p+1-2j)}$$.
• Otherwise, $$\sigma^{(j)}\otimes\sigma^{(\ell)} \cong \sigma^{(r)}\oplus\sigma^{(s)}$$ for suitable $r,s$.

Explicit coupled bases for small $p$ coincide with relabelings of the familiar Bell states.

4. Entanglement and Ultrametric Features

A state in a $p$-adic tensor-product space is entangled iff it cannot be written as $x\otimes y$. The projective norm and orthonormal basis facilitate definitions of analogues of density matrices, partial trace, and Schmidt rank. The ultrametric norm precludes the cancellation effects found in the complex case—interference is "maximal," with $|x+y|_p \leq \max(|x|_p,|y|_p)$. The structure of entangled states is distinctive: Proposition 5.1 of (Svampa et al., 20 Jan 2026) demonstrates that every one-dimensional subrepresentation corresponds to a maximally entangled Bell-type state, while higher-dimensional stable subspaces are separable according to the Peres–Horodecki (PPT) criterion. This stands in contrast to the standard complex setting, where entanglement may occur in multi-dimensional subspaces.

5. $p$-adic Quantum Gates and Universal Computation

Local $p$-adic gates are elements of $GL_{2^n}(Z_p)$ preserving the sup-norm and the $p$-adic inner product. For a single-qubit, gates are $2\times2$ matrices with $Z_p$ entries and determinant in $Z_p^\times$. Composition of qubit gates is realized as matrix multiplication and, in the quaternionic formulation, corresponds exactly to quaternion multiplication—thus gate composition reflects group composition in the noncommutative $p$-adic quaternion algebra $H(A,B)$ over $Q_p$. Haar measure on the analytic group of unit quaternions $U(H)$ provides the natural probability measure for random gate ensembles (Aniello et al., 2023).

For two-qubit gates, tensor product constructions yield $4\times4$ matrices in $GL_4(Z_p)$, and in the $p=3$ case, $G_3=SO(3)_3\bmod3$ admits four inequivalent 4-dimensional irreps which generate gate sets dense in $O(4)$, proving universality. These gates split into tensor-factorizing and genuinely entangling cosets; adjoining further ancilla qubits and encoding complex gates as real orthogonal matrices supplies an approximately universal set for any $n$ (Svampa et al., 20 Jan 2026).

6. Extensions: Adelic Constructions and Multi-Qubit Systems

The adelic framework generalizes $p$-adic qubit composition to global fields $F$ and their ring of adeles $A$. The global $n$-qubit space $H_A^{(n)}=A\otimes_{\Q} H_{\Q}^{(n)}\cong A^{2^n}$ integrates local $p$-adic and Archimedean factors, with constrained support conditions ensuring almost all local coordinates lie in maximal compact subrings (Weng, 2021). Adelic gates form the product $$K=\prod_{v\,\mathrm{finite}}GL_{2^n}(O_v) \times \prod_{o\,|\,\infty}U(2^n)$$, where each local gate can be embedded as the $p$-component of an adele. Tensor-product rules and norm-preservation are formally identical to the purely local theory.

7. Structural Comparison and Implications for Quantum Information Theory

The $p$-adic tensor product of Hilbert spaces deviates from complex quantum information in several major aspects. The ultrametric norm, manifest as a max-norm, enforces non-Archimedean orthogonality (Birkhoff–James) and blocks standard cancellation-based interference. Subspace structure and closure must be handled carefully, as direct-sum decompositions do not necessarily behave as expected under the ultrametric topology. Nonetheless, the algebraic and representation-theoretic underpinnings permit rigorous analogues of composite systems, entangled states, and universal gate sets, laying the foundations for the study of $p$-adic quantum information and possibly cryptographic applications. The explicit isomorphism between completed $p$-adic tensor products and Hilbert–Schmidt operator classes (Aniello et al., 8 Oct 2025) suggests potentially tractable operator-theoretic characterizations of $p$-adic quantum circuits.

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Table: Key Constructs in $p$-adic Qubit Composition

Construct	Complex Case Analog	$p$-adic Feature
State space	$\C^2$	$\Q_p^2$, sup-norm
Composite system	Tensor product Hilbert space	$p$-adic tensor product, max-norm
Gate group	$U(2)$, $U(2^n)$	$GL_{2^n}(Z_p)$, $SO(3)_p$, $U(H)$
Entanglement criterion	Schmidt rank	Non-factorizable in tensor product
Universal gate set	Dense subset of $SU(4)$	Dense subset of $O(4)$ ($p=3$)
Haar measure (gate ensemble)	Unitary group measure	Product measure on unit quaternions

A plausible implication is that $p$-adic quantum information theory can exploit ultrametric phenomena for novel protocols, including error correction adapted to non-Archimedean metrics and representation-theoretic gate synthesis in finite quotient rotation groups. The non-Archimedean algebraic structure offers rich future directions in arithmetic quantum information, cryptography, and $p$-adic harmonic analysis.