Criteria to detect genuine multipartite entanglement using spin measurements

Abstract

We derive conditions in the form of inequalities to detect the genuine N -partite entanglement of N 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 ∆Jx∆Jy ≥ | Jz |/2. We also extend previous work to derive spin-variance inequalities that certify the full tripartite inseparability or genuine multi-partite entanglement among systems with fixed spin J, as in Greenberger-Horne-Zeilinger (GHZ) states and W states where J = 1/2. These inequalities are derived from the planar spin-uncertainty relation (∆Jx) 2 + (∆Jy) 2 ≥ CJ where CJ is a constant for each J. 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.

Detection of genuine multipartite entanglement based on local sum uncertainty relations

“…When we choose spin observables as the observables A, B, and C, the criteria in Theorem 3 require only the statistics of a set of observables. In this sense, it is state independent, which is similar to [31]. In order to compare Theorem 3 with criterion 1 in [31], we consider the sum of ∆ 2 u and ∆ 2 v where…”

“…In this sense, it is state independent, which is similar to [31]. In order to compare Theorem 3 with criterion 1 in [31], we consider the sum of ∆ 2 u and ∆ 2 v where…”

“…The converse negative process is the criterion of detecting GME. When we choose spin observables, the criterion is better than that in [31]. Third, we use the criterion detect 3-qutrit state and noisy W state for n-qubit system (n = 3, 4, 5, 6) and find that the criterion is strong than the exiting ones [24,32].…”

Non-gaussian Entanglement Swapping between Three-Mode Spontaneous Parametric Down Conversion and Three Qubits

“…Interestingly, the concepts of gaussian and nongaussian entanglement can be related with the two main representatives of tripartite qubit entanglement, the W and GHZ states. The W -entanglement is gaussian, since can be detected by a gaussian witness [25], while GHZentanglement is nongaussian, since we can for instance find a state that contains no entanglement and yet has the same means and covariances on the spin variables as the GHZ state:…”

Hierarchy of Nonlinear Entanglement Dynamics for Continuous Variables

“…[14,15]. As an extension of these works, a series of entanglement criteria for multimode Gaussian states were proposed [16][17][18][19][20][21]. Gaussian states are out of the ordinary because of their simple generation and manipulation, and their versatility in quantum information protocols [10,22].…”