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.

中文翻译：

---

正常排序中的算子转置及其在量化纠缠中的应用

部分转置是量化纠缠的重要操作。在这项研究中，研究了任何单个（双模式）算子的（部分）转置。使用Fock基展开，发现任意算子的转置算子可以通过替换${\mathrm{一种}}^{†}(\mathrm{一种})$ 和 $\mathrm{一种}({\mathrm{一种}}^{†})$, 而不是正常订购表格中的 c 编号。还研究了位移算子和Wigner算子的转置，从中构建了密度算子和转置密度算子之间的Wigner函数、特征函数和协方差矩阵等平均值的关系。这些观察结果可以进一步扩展到多模式情况。在应用方面，考虑了二模挤压算子的部分转置和二模挤压真空通过激光通道的纠缠。