Natural Qubit Algebra: clarification of the Clifford boundary and new non-embeddability theorem

Abstract

We introduce Natural Qubit Algebra (NQA), a compact real operator calculus for qubit systems based on a $2\times2$ block alphabet $\{I,X,Z,W\}\subset\mathrm{Mat}(2,\mathbb{R})$ and tensor-word representations. The resulting multiplication law induces a canonical $(\mathbb{Z}_2)^{2m}$-grading with a bicharacter that controls commutation signs, placing the framework naturally within the theory of color-graded and Clifford-type algebras. Within this language, we provide: (i) an explicit real Clifford normal form for two-qubit operators via the identification $\mathrm{Mat}(4,\mathbb{R})\cong\mathrm{Cl}(2,2;\mathbb{R})$; (ii) a purely algebraic reformulation of the Bell--CHSH scenario, where the quantum violation is expressed as a spectral non-embeddability of a noncommutative spinor algebra into any commutative Kolmogorov algebra; and (iii) compact factored representations of the Bernstein--Vazirani and Grover phase oracles, showing that both Clifford and non-Clifford examples can admit similarly structured symbolic descriptions. We clarify that Grover's iterate remains outside the Clifford group due to its continuous spectral rotation, consistent with the Gottesman--Knill theorem, while retaining a compact tensor-block form in NQA. The framework isolates spectral, algebraic, and syntactic aspects of operator structure, providing a graded operator language compatible with standard quantum mechanics.