Operator Transpose within Normal Ordering and its Applications for Quantifying Entanglement

Partial transpose is an important operation for quantifying entanglement. In this study, the (partial) transpose of any single (two-mode) operator is investigated. Using the Fock basis expansion, it is found that the transposed operator of an arbitrary operator can be obtained by replacing $a^{†}(a)$ with $a(a^{†})$, rather than the c-number within the normal ordering form. The transpose of the displacement and Wigner operators is also investigated, from which the relation of the Wigner function, characteristics function, and average values such as covariance matrix is constructed between the density operator and transposed density operator. These observations can be further extended to multi-mode cases. As for the application, partial transpose of the two-mode squeezed operator and the entanglement of the two-mode squeezed vacuum through a laser channel is considered.