Bures distance as a measure of entanglement for symmetric two-mode Gaussian states

TL;DR

This paper introduces a new way to measure entanglement in symmetric two-mode Gaussian states using the Bures distance, simplifying calculations and aligning with known entanglement measures.

Contribution

It provides a simplified method to evaluate Gaussian entanglement via Bures distance, linking it to the symplectic eigenvalues and entanglement of formation.

Findings

Bures distance-based entanglement measure depends only on the smallest symplectic eigenvalue.

The measure aligns with the exact entanglement of formation for symmetric states.

Simplified minimization procedure using properties of Uhlmann fidelity.

Abstract

We evaluate a Gaussian entanglement measure for a symmetric two-mode Gaussian state of the quantum electromagnetic field in terms of its Bures distance to the set of all separable Gaussian states. The required minimization procedure was considerably simplified by using the remarkable properties of the Uhlmann fidelity as well as the standard form II of the covariance matrix of a symmetric state. Our result for the Gaussian degree of entanglement measured by the Bures distance depends only on the smallest symplectic eigenvalue of the covariance matrix of the partially transposed density operator. It is thus consistent to the exact expression of the entanglement of formation for symmetric two-mode Gaussian states. This non-trivial agreement is specific to the Bures metric.