Algebraic Criteria for Entanglement in Multipartite Systems

Abstract

Quantum computing depends heavily on quantum entanglement. It has been known that geometric models for correlated two-state quantum systems (qubits) can be developed using geometric algebra. This suggests that entanglement may be given a purely algebraic description without resort to any particular representation on Hilbert spaces. In the case of the Clifford algebra, for example, the states are not simply operands in a Hilbert space representation of the algebra but they are considered as embedded within the Clifford algebra itself. In other words the space of states sits inside the algebra. This Clifford-algebraic substructure is a minimal left ideal of the algebra. This fact naturally poses the question of whether or not the description of entanglement in multipartite systems can be generalized to algebras possessing one-sided ideal structure. By making tensor products of algebras and their minimal one-sided ideals we propose an algebraic criteria for characterizing entanglement in multipartite systems without resort to any representation on Hilbert spaces.