Tomita-Takesaki theory and quantum concurrence

The quantum entanglement measure of concurrence is shown to be directly calculable from a Tomita- Takesaki modular operator framework constructed from the local von Neumann algebras of observables for two quantum systems. Specifically, the Tomita-Takesaki modular conjugation operator J that links two separate systems with respect to their von Neumann algebras is related to the quantum concurrence C of a pure bi-variate entangled state composed from these systems. This concurrence relation provides a direct physical meaning to J as both a symmetry operator and a quantitative measure of entanglement. This procedure is then demonstrated for a supersymmetric quantum mechanical system and a real scalar field interacting with two entangled spin-12 Unruh-DeWitt qubit detectors. For the latter system, the concurrence result is shown to be consistent with some known results on the Bell-CHSH inequality for such a system.