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  • Open Access

Revisiting Gaussian genuine entanglement witnesses with modern software

E. Shchukin* and P. van Loock†

  • *Contact author: shchukin@uni-mainz.de
  • †Contact author: loock@uni-mainz.de
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Phys. Rev. Research 8, 023124 – Published 5 May, 2026

DOI: https://doi.org/10.1103/drmr-qf2q

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 nonphysical 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.

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Phys. Rev. Research 8, 023124– Published 5 May, 2026
  • Received 16 September 2025
  • Accepted 6 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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