  The modular operator approach of Tomita-Takesaki to von Neumann algebras is
elucidated in the algebraic structure of certain supersymmetric 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 supersymmetric two dimensional system is given by the modular
conjugation operator. Furthermore, the entanglement of formation for these
supersymmetric 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 supersymmetric Jaynes Cummings
Model.
