Operator transpose within normal ordering and its applications for quantifying entanglement

TL;DR

This paper investigates the partial transpose operation on quantum operators within normal ordering, providing new methods to analyze entanglement and its quantification, especially for two-mode systems and their transformations.

Contribution

It introduces a novel approach to compute the transpose of operators in normal ordering, facilitating entanglement analysis in quantum systems.

Findings

Derived relations between Wigner functions and transposed density operators.

Extended the operator transpose method to multi-mode systems.

Analyzed entanglement of two-mode squeezed vacuum via partial transpose.

Abstract

Partial transpose is an important operation for quantifying the entanglement, here we study the (partial) transpose of any single (two-mode) operators. Using the Fock-basis expansion, it is found that the transposed operator of an arbitrary operator can be obtained by replacement of a^{{\dag}}(a) by a(a^{{\dag}}) instead of c-number within normal ordering form. The transpose of displacement operator and Wigner operator are studied, from which the relation of Wigner function, characteristics function and average values such as covariance matrix are constructed between density operator and transposed density operator. These observations can be further extended to multi-mode cases. As applications, the partial transpose of two-mode squeezed operator and the entanglement of two-mode squeezed vacuum through a laser channel are considered.