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 Cliffordalgebraic 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.