The modular operator approach of Tomita–Takesaki to von Neumann algebras is elucidated in the algebraic structure of certain supersymmetric (SUSY) quantum mechanical systems. A von Neumann algebra is constructed from the operators of the system. An explicit operator characterizing the dual infinite degeneracy structure of a SUSY two dimensional system is given by the modular conjugation operator. Furthermore, the entanglement of formation for these SUSY systems using concurrence is shown to be related to the expectation value of the modular conjugation operator in an entangled bi-partite supermultiplet state thus providing a direct physical meaning to this anti-unitary, anti-linear operator as a quantitative measure of entanglement. Finally, the theory is applied to the case of two-dimensional Dirac fermions, as is found in graphene, and a SUSY Jaynes Cummings model.

在某些超对称 (SUSY) 量子力学系统的代数结构中阐明了 Tomita-Takesaki 对冯诺依曼代数的模算子方法。冯诺依曼代数由系统的算子构成。模共轭算子给出了表征 SUSY 二维系统的对偶无限简并结构的显式算子。此外，这些使用并发的 SUSY 系统的形成纠缠被证明与纠缠二分超重态中模共轭算子的期望值有关，从而为这种反酉、反线性算子提供了直接的物理意义作为纠缠的定量度量。最后，将该理论应用于二维狄拉克费米子的情况，如在石墨烯中发现的那样，