# 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/en/arxiv-id/2512.07902v2/

## Key points

- Kagwe A. Muchane explores the connection between Pauli and Clifford operators in $n$-qubit quantum computation
- Real algebraic framework based on the tensor product $C\ell_{2,0}(\mathbb{R})^{\otimes n}$
- Bivector $J = e_{12}$ plays a crucial role by providing complex structure on the $J$-closure of a minimal left ideal through right multiplication
- Peirce decomposition categorizes the algebra into sector blocks defined by primitive idempotents
- Nilpotent elements facilitate transitions between sectors
- Quantum states depicted as equivalence classes modulo the left annihilator, highlighting quotient description inherent in minimal left ideal
- Canonical stabilizer mapping shows that the $n$-qubit computational basis state $|0\cdots 0\rangle$ can be represented as a tensor product of idempotents
- Structural choice leads to compatibility law stable under geometric product for $n$ qubits and aligns symbolic Clifford multiplication with unitary evolution within Hilbert space
- Example of Grover search included in appendix showcasing leveraging real algebraic structures for advancing quantum information processing methodologies

## Summary

In "The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information," Kagwe A. Muchane explores the connection between Pauli and Clifford operators in $n$-qubit quantum computation using a real algebraic framework based on the tensor product $C\ell_{2,0}(\mathbb{R})^{\otimes n}$. The bivector $J = e_{12}$ plays a crucial role in this setting by providing the complex structure on the $J$-closure of a minimal left ideal through right multiplication. By delving into the Peirce decomposition, Muchane categorizes the algebra into sector blocks defined by primitive idempotents and showcases how nilpotent elements facilitate transitions between these sectors. Furthermore, he depicts quantum states as equivalence classes modulo the left annihilator, highlighting the quotient description inherent in the minimal left ideal. Through a canonical stabilizer mapping, Muchane demonstrates that the $n$-qubit computational basis state $|0\cdots 0\rangle$ can be represented as a tensor product of these idempotents. This structural choice leads to a compatibility law that remains stable under geometric product for $n$ qubits and aligns symbolic Clifford multiplication with unitary evolution within the Hilbert space. In addition to presenting an expanded framework, Muchane includes an example of Grover search in the appendix and offers valuable insights into leveraging real algebraic structures for advancing quantum information processing methodologies.