Geometric characterization of separability and entanglement in pure Gaussian states by single-mode unitary operations

TL;DR

This paper introduces a geometric method to quantify entanglement in pure Gaussian states using single-mode unitary operations, providing a new entanglement measure that is experimentally accessible.

Contribution

It extends a formalism from finite-dimensional systems to continuous variables, defining a novel entanglement monotone based on minimum Euclidean distance under local symplectic transformations.

Findings

Entanglement can be quantified by a minimum Euclidean distance in phase space.

The proposed measure is equivalent to the entropy of entanglement.

The measure can be directly measured using linear optical schemes.

Abstract

We present a geometric approach to the characterization of separability and entanglement in pure Gaussian states of an arbitrary number of modes. The analysis is performed adapting to continuous variables a formalism based on single subsystem unitary transformations that has been recently introduced to characterize separability and entanglement in pure states of qubits and qutrits [arXiv:0706.1561]. In analogy with the finite-dimensional case, we demonstrate that the $1 \times M$ bipartite entanglement of a multimode pure Gaussian state can be quantified by the minimum squared Euclidean distance between the state itself and the set of states obtained by transforming it via suitable local symplectic (unitary) operations. This minimum distance, corresponding to a, uniquely determined, extremal local operation, defines a novel entanglement monotone equivalent to the entropy of entanglement, and amenable to direct experimental measurement with linear optical schemes.