We revisit the Bell--CHSH scenario for two spin-$\tfrac{1}{2}$ particles and isolate the precise algebraic origin of the Bell contradiction. On the quantum side, spin-$\tfrac{1}{2}$ is described by a noncommutative spinor (Clifford) algebra acting on the Hilbert space of two spin-$\tfrac{1}{2}$ particles, with the singlet state yielding the usual correlation $E(a,b) = -\,a\cdot b$ and Tsirelson's bound $2\sqrt{2}$. On the classical side, the standard Bell assumptions amount to describing all measurement outcomes as $\{\pm1\}$-valued random variables on a single Kolmogorov probability space, i.e.\ elements of a commutative algebra $\mathcal{C}(Λ)$. We show that there is no representation of the spinor algebra of spin-$\tfrac{1}{2}$ (with its singlet state and locality structure) into any such commutative Kolmogorov algebra that preserves the $\{\pm1\}$ spectra of local spin components and the singlet correlations entering the CHSH expression, under the standard Bell assumptions of locality (factorization) and measurement independence. In this sense, the Bell--CHSH contradiction is exhibited as an algebraic mismatch between a noncommutative spinor/Clifford description of spin and the classical assumption of a single global Kolmogorov space supporting all outcomes. In the language of quantum probability, this is a C$^*$-algebraic reformulation of the known fact that the singlet correlations admit no local hidden-variable model with jointly distributed outcomes on one probability space. We also give an explicit realization of the same spinor structure within the author's Quantum Index Algebra (QIA) framework, where locality appears as disjoint index slots and the singlet state as a simple index cocycle.

quant-pharXiv:2512.11476Dec 15, 20250 citations

arXiv:2001.00656 · Advances in Applied Clifford Algebras

Two-State Quantum Systems Revisited: A Clifford Algebra Approach

Pedro Amao, Hernán Castillo

Published Mar 8, 2021• 4 citations

arXiv HTML PDF

DOI: 10.1007/s00006-020-01116-1

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quant-pharXiv:2512.11476Dec 15, 20250 citations

We revisit the Pauli-Clifford connection to introduce a real, grade-preserving algebraic framework for

n

-qubit quantum computation based on the tensor product

C\ell_{2,0}(\mathbb{R})^{\otimes n}

. In this setting, the bivector

J = e_{12}

satisfies

J^{2} = -1

and supplies the complex structure on the

J

-closure of a minimal left ideal via right multiplication, while Pauli operations arise as left actions of Clifford elements. The Peirce decomposition organizes the algebra into sector blocks determined by primitive idempotents, with nilpotent elements generating transitions between sectors. Quantum states are represented as equivalence classes modulo the left annihilator, exhibiting the quotient description underlying the minimal left ideal. Adopting a canonical stabilizer mapping, the

n

-qubit computational basis state

|0\cdots 0\rangle

is given natively by a tensor product of these idempotents. This structural choice leads to a compatibility law that is stable under the geometric product for

n

qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.

We revisit the Bell--CHSH scenario for two spin-

\tfrac{1}{2}

particles and isolate the precise algebraic origin of the Bell contradiction. On the quantum side, spin-

\tfrac{1}{2}

is described by a noncommutative spinor (Clifford) algebra acting on the Hilbert space of two spin-

\tfrac{1}{2}

particles, with the singlet state yielding the usual correlation

E(a,b) = -\,a\cdot b

and Tsirelson's bound

2\sqrt{2}

. On the classical side, the standard Bell assumptions amount to describing all measurement outcomes as

\{\pm1\}

-valued random variables on a single Kolmogorov probability space, i.e.\ elements of a commutative algebra

\mathcal{C}(Λ)

. We show that there is no representation of the spinor algebra of spin-

\tfrac{1}{2}

(with its singlet state and locality structure) into any such commutative Kolmogorov algebra that preserves the

\{\pm1\}

spectra of local spin components and the singlet correlations entering the CHSH expression, under the standard Bell assumptions of locality (factorization) and measurement independence. In this sense, the Bell--CHSH contradiction is exhibited as an algebraic mismatch between a noncommutative spinor/Clifford description of spin and the classical assumption of a single global Kolmogorov space supporting all outcomes. In the language of quantum probability, this is a C

^*

-algebraic reformulation of the known fact that the singlet correlations admit no local hidden-variable model with jointly distributed outcomes on one probability space. We also give an explicit realization of the same spinor structure within the author's Quantum Index Algebra (QIA) framework, where locality appears as disjoint index slots and the singlet state as a simple index cocycle.