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arXiv:1407.7248 · Physical Review A

Systematic construction of genuine-multipartite-entanglement criteria in continuous-variable systems using uncertainty relations

Fabricio Toscano, A. Saboia, Ardiley Torres Avelar, Stephen Patrick Walborn

Published Nov 11, 2015• 23 citations
arXivHTMLPDF

Abstract

A general procedure to construct criteria for identifying genuine multipartite continuous-variable entanglement is presented. It relies on the definition of adequate global operators describing the multipartite system, the positive partial transpose criterion of separability, and quantum-mechanical uncertainty relations. As a consequence, each criterion encountered consists of a single inequality that is nicely computable and experimentally feasible. Violation of the inequality is a sufficient condition for genuine multipartite entanglement. Additionally, we show that the previous work of van Loock and Furusawa [P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003)] is a special case of our result.

DOI
10.1103/physreva.92.052316

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A general procedure to construct criteria for identifying genuine multipartite continuous-variable entanglement is presented. It relies on the definition of adequate global operators describing the multipartite system, the positive partial transpose criterion of separability, and quantum-mechanical uncertainty relations. As a consequence, each criterion encountered consists of a single inequality that is nicely computable and experimentally feasible. Violation of the inequality is a sufficient condition for genuine multipartite entanglement. Additionally, we show that the previous work of van Loock and Furusawa [P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003)] is a special case of our result.
We derive conditions in the form of inequalities to detect the genuine NNN-partite entanglement of NNN systems. The inequalities are expressed in terms of variances of spin operators and can be tested by local spin measurements performed on the individual systems. Violation of the inequalities is sufficient (but not necessary) to certify the multipartite entanglement and occurs when a type of spin squeezing is created. The inequalities are similar to those derived for continuous-variable systems, but instead are based on the Heisenberg spin-uncertainty relation $\mathrm{\ensuremath{\Delta}}{J}_{x}\mathrm{\ensuremath{\Delta}}{J}_{y}\ensuremath{\ge}|\ensuremath{\langle}{J}_{z}\ensuremath{\rangle}|/2$. We also extend previous work to derive spin-variance inequalities that certify the full tripartite inseparability or genuine multipartite entanglement among systems with fixed spin JJJ, as in Greenberger--Horne--Zeilinger (GHZ) states and W states where J=1/2J=1/2J=1/2. These inequalities are derived from the planar spin-uncertainty relation ${(\mathrm{\ensuremath{\Delta}}{J}_{x})}^{2}+{(\mathrm{\ensuremath{\Delta}}{J}_{y})}^{2}\ensuremath{\ge}{C}_{J}$ where CJ{C}_{J}CJ​ is a constant for each JJJ. Finally, it is shown how the inequalities detect multipartite entanglement based on Stokes operators. We illustrate with experiments that create entanglement shared among separated atomic ensembles, polarization-entangled optical modes, and the clouds of atoms of an expanding spin-squeezed Bose-Einstein condensate. For each example, we give a criterion to certify the mutual entanglement.
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