# The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information

**Authors:** Kagwe A. Muchane  
**arXiv:** [2512.07902](https://arxiv.org/abs/2512.07902)  
**Date:** 2025-12-05  
**Summary page:** https://www.summarizepaper.com/fr/arxiv-id/2512.07902v1/

## Abstract

We revisit the Pauli-Clifford connection to introduce a real, grade-preserving algebraic framework for $N$-qubit quantum computation based on the tensor product structure $C\ell_{2,0}(\mathbb{R})^{\otimes N}$. In this setting the bivector $J = e_{12}$ satisfies $J^{2} = -1$ and supplies the complex structure on a minimal left ideal via right-multiplication, while Pauli operations arise as left actions of suitable Clifford elements. Adopting a canonical stabilizer mapping, the $N$-qubit computational basis state $|0\cdots 0\rangle$ is represented natively by a tensor product of real algebraic idempotents. This structural choice leads to a State-Operator Clifford Compatibility law that is stable under the geometric product for $N$ qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.