Gaussian intrinsic entanglement: An entanglement quantifier based on secret correlations

TL;DR

This paper introduces Gaussian intrinsic entanglement (GIE), a new entanglement measure based on secret correlations, demonstrating its properties, calculation methods, and relation to existing entanglement quantifiers for Gaussian states.

Contribution

The paper defines GIE, proves its key properties, computes it for specific states, and extends the concept to certain non-Gaussian states, linking it to Gaussian Rényi-2 entanglement.

Findings

GIE vanishes on all Gaussian separable states.

GIE is non-increasing under Gaussian LOCC.

GIE equals Gaussian Rényi-2 entanglement for pure states.

Abstract

Intrinsic entanglement (IE) is a quantity which aims at quantifying bipartite entanglement carried by a quantum state as an optimal amount of the intrinsic information that can be extracted from the state by measurement. We investigate in detail the properties of a Gaussian version of IE, the so-called Gaussian intrinsic entanglement (GIE). We show explicitly how GIE simplifies to the mutual information of a distribution of outcomes of measurements on a conditional state obtained by a measurement on a purifying subsystem of the analyzed state, which is first minimized over all measurements on the purifying subsystem and then maximized over all measurements on the conditional state. By constructing for any separable Gaussian state a purification and a measurement on the purifying subsystem which projects the purification onto a product state, we prove that GIE vanishes on all Gaussian separable states. Via realization of quantum operations by teleportation, we further show that GIE is non-increasing under Gaussian local trace-preserving operations and classical communication. For pure Gaussian states and a reduction of the continuous-variable GHZ state, we calculate GIE analytically and we show that it is always equal to the Gaussian R\'{e}nyi-2 entanglement. We also extend the analysis of IE to a non-Gaussian case by deriving an analytical lower bound on IE for a particular form of the non-Gaussian continuous-variable Werner state. Our results indicate that mapping of entanglement onto intrinsic information is capable of transmitting also quantitative properties of entanglement and that this property can be used for introduction of a quantifier of Gaussian entanglement which is a compromise between computable and physically meaningful entanglement quantifiers.