# Einstein-Podolsky-Rosen-like separability indicators for two-mode Gaussian states

## TL;DR

This paper introduces new EPR-like variances as indicators for the separability of two-mode Gaussian states, establishing their equivalence with the well-known PPT criterion and enhancing understanding of quantum entanglement detection.

## Contribution

It develops novel EPR-like separability indicators based on variances, demonstrating their equivalence with the PPT criterion for two-mode Gaussian states.

## Key findings

- EPR-like variances serve as effective separability indicators.
- The EPR-like approach is equivalent to the PPT criterion.
- The methods quantify the maximum EPR correlations achievable.

## Abstract

We investigate the separability of the two-mode Gaussian states by using the variances of a pair of Einstein-Podolsky-Rosen (EPR)-like observables. Our starting point is inspired by the general necessary condition of separability introduced by Duan {\em et al.} [Phys. Rev. Lett. {\bf 84}, 2722 (2000)]. We evaluate the minima of the normalized forms of both the product and sum of such variances, as well as that of a regularized sum. Making use of Simon's separability criterion, which is based on the condition of positivity of the partial transpose (PPT) of the density matrix [Phys. Rev. Lett. {\bf 84}, 2726 (2000)], we prove that these minima are separability indicators in their own right. They appear to quantify the greatest amount of EPR-like correlations that can be created in a two-mode Gaussian state by means of local operations. Furthermore, we reconsider the EPR-like approach to the separability of two-mode Gaussian states which was developed by Duan {\em et al.} with no reference to the PPT condition. By optimizing the regularized form of their EPR-like uncertainty sum, we derive a separability indicator for any two-mode Gaussian state. We prove that the corresponding EPR-like condition of separability is manifestly equivalent to Simon's PPT one. The consistency of these two distinct approaches (EPR-like and PPT) affords a better understanding of the examined separability problem, whose explicit solution found long ago by Simon covers all situations of interest.