Revisiting Gaussian genuine entanglement witnesses with modern software

TL;DR

This paper develops convex optimization methods to reconstruct physical covariance matrices from measurements in Gaussian quantum systems and tests their entanglement properties, introducing analytical tools and applying them to complex states.

Contribution

It introduces efficient convex optimization approaches for verifying entanglement in Gaussian states, including analytical solutions and explicit formulas for entanglement witnesses.

Findings

Reconstructed physical covariance matrices from non-physical measurements.

Verified various forms of separability and entanglement in Gaussian states.

Derived an explicit analytical expression for the symplectic trace as an entanglement witness.

Abstract

Continuous-variable Gaussian entanglement is an attractive notion, both as a fundamental concept in quantum information theory, based on the well-established Gaussian formalism for phase-space variables, and as a practical resource in quantum technology, exploiting in particular, unconditional room-temperature squeezed-light quantum optics. The readily available high level of scalability, however, is accompanied by an increased theoretical complexity when the multipartite entanglement of a growing number of optical modes is considered. For such systems, we present several approaches to reconstruct the most probable physical covariance matrix from a measured non-physical one and then test the reconstructed matrix for different kinds of separability (factorizability, concrete partite separability or biseparability) even in the presence of measurement errors. All these approaches are based on formulating the desired properties (physicality or separability) as convex optimization problems, which can be efficiently solved with modern optimization solvers, even when the system grows. To every optimization problem we construct the corresponding dual problem used to verify the optimality of the solution. Besides this numerical part of work, we derive an explicit analytical expression for the symplectic trace of a positive definite matrix, which can serve as a simple witness of an entanglement witness, and extend it for positive semidefinite matrices. In addition, we show that in some cases our optimization problems can be solved analytically. As an application of our analytical approach, we consider small instances of bound entangled or genuine multipartite entangled Gaussian states, including some examples from the literature that were treated only numerically, and a family of non-Gaussian states.