Generalized Entangled Wigner Operator for Unifying Three Quantization Schemes of Entangled Systems

Abstract

For entangled systems since wave function for single particle is not physical, marginal distributions of the Wigner function for entangled states are only meaningful in the entangled state representation, we propose generalized Weyl quantization scheme which relies on the generalized entangled Wigner operator ${\Omega }_{k}\left (\sigma ,\gamma \right )$ with a real k-parameter and which can unify $\mathfrak {P}-$ordering, $\mathfrak {X}-$ ordering and Weyl ordering of operators in k = 1, −1, 0 respectively, we also find the mutual transformations among the integration kernel of $\mathfrak {P}-$ordering, $\mathfrak {X}-$ ordering and generalized Weyl quantization schemes. The mutual transformations provides us with a new approach for deriving Wigner function of entangled quantum states. The $\mathfrak {P}-$ordered and $\mathfrak {X}-$ordered form of ${\Omega }_{k}\left (\sigma ,\gamma \right )$ are also derived which helps to put operators into their $\mathfrak {P}-$ordering and $\mathfrak {X}-$ordering respectively.