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 bivariate 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-$\frac{1}{2}$ 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.

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富田竹崎理论与量子并发


量子纠缠并发度量被证明可以直接从 Tomita-Takesaki 模算子框架计算，该框架是根据两个量子系统的可观测量的局部冯诺依曼代数构建的。具体来说，Tomita-Takesaki 模共轭算子 $J$ 将两个独立系统的冯诺依曼代数联系起来与量子并发有关 $C$ 由这些系统组成的纯二元纠缠态。这种并发关系提供了直接的物理意义 $J$ 作为对称算子和纠缠的定量度量。然后针对超对称量子力学系统和与两个纠缠自旋相互作用的实标量场演示了该过程。 $\frac{1}{2}$ Unruh-DeWitt 量子位探测器。对于后一个系统，并发结果与此类系统的 Bell-CHSH 不等式的一些已知结果一致。