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arXiv:quant-ph/0601038 · Physical Review A

Generalized spin-squeezing inequalities in N -qubit systems: Theory and experiment

Jarosław K. Korbicz, Otfried Gühne, Maciej Lewenstein, Hartmut Häffner, Christian F. Roos, R. Blatt

Published Nov 14, 2006• 96 citations
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Abstract

We present detailed derivations, various improvements and application to concrete experimental data of spin squeezing inequalities formulated recently by some of us [Phys. Rev. Lett. 95, 120502 (2005)]. These inequalities generalize the concept of the spin squeezing parameter and provide necessary and sufficient conditions for genuine two-, or three-qubit entanglement for symmetric states, and sufficient condition for general $N$-qubit states. We apply our method to theoretical study of Dicke states, and, in particular, to $\mathrm{W}$ states of $N$ qubits. Then, we analyze the recently experimentally generated seven- and eight-ion $\mathrm{W}$ states [Nature 438, 643 (2005)]. we also present some details concerning this experiment. Finally, we improve criteria for detection of genuine tripartite entanglement based on entanglement witnesses.

DOI
10.1103/physreva.74.052319

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We present detailed derivations, various improvements and application to concrete experimental data of spin squeezing inequalities formulated recently by some of us [Phys. Rev. Lett. 95, 120502 (2005)]. These inequalities generalize the concept of the spin squeezing parameter and provide necessary and sufficient conditions for genuine two-, or three-qubit entanglement for symmetric states, and sufficient condition for general NNN-qubit states. We apply our method to theoretical study of Dicke states, and, in particular, to W\mathrm{W}W states of NNN qubits. Then, we analyze the recently experimentally generated seven- and eight-ion W\mathrm{W}W states [Nature 438, 643 (2005)]. we also present some details concerning this experiment. Finally, we improve criteria for detection of genuine tripartite entanglement based on entanglement witnesses.
What is the relation between spin squeezing and entanglement? To clarify this, we derive the full set of generalized-spin-squeezing inequalities for the detection of entanglement. These are inequalities for the mean values and variances of the collective angular-momentum components Jk{J}_{k}Jk​. They can be used for the experimental detection of entanglement in a system of spin-12\frac{1}{2}21​ particles in which the spins cannot be individually addressed. We present various sets of inequalities that can detect all entangled states that can be detected based on the knowledge of (i) the mean values and variances of Jk{J}_{k}Jk​ in three orthogonal directions, or (ii) the variances of Jk{J}_{k}Jk​ in three orthogonal directions, or (iii) the mean values of Jk2{J}_{k}^{2}Jk2​ in three orthogonal directions, or (iv) the mean values and variances of Jk{J}_{k}Jk​ in arbitrary directions. We compare our inequalities to known spin-squeezing entanglement criteria and discuss to which extent spin squeezing is related to entanglement in the reduced two-qubit states. Finally, we apply our criteria for the detection of entanglement in spin models, showing that they can be used to detect bound entanglement in these systems.
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.
A complete set of generalized spin-squeezing inequalities is derived for an ensemble of particles with an arbitrary spin. Our conditions are formulated with the first and second moments of the collective angular momentum coordinates. A method for mapping the spin-squeezing inequalities for spin-12\frac{1}{2}21​ particles to entanglement conditions for spin-jjj particles is also presented. We apply our mapping to obtain a generalization of the original spin-squeezing inequality to higher spins. We show that, for large particle numbers, a spin-squeezing parameter for entanglement detection based on one of our inequalities is strictly stronger than the original spin-squeezing parameter defined in S\o{}rensen et al. [Nature (London) 409, 63 (2001)]. We present a coordinate system independent form of our inequalities that contains, besides the correlation and covariance tensors of the collective angular momentum operators, the nematic tensor appearing in the theory of spin nematics. Finally, we discuss how to measure the quantities appearing in our inequalities in experiments.
Pairwise entanglement properties of a symmetric multi-qubit system are analyzed through a complete set of two-qubit local invariants. Collective features of entanglement, such as spin squeezing, are expressed in terms of invariants and a classification scheme for pairwise entanglement is proposed. The invariant criteria given here are shown to be related to the recently proposed (Phys. Rev. Lett.95, 120502 (2005)) generalized spin squeezing inequalities for pairwise entanglement in symmetric multi-qubit states.
We show that spin-squeezing criteria commonly used for entanglement detection can be erroneous if the probe is not symmetric. We then derive a lower bound on squeezing for separable states in spin systems probed asymmetrically. Using this we further develop a procedure that allows us to verify the degree of entanglement of a quantum state in the spin system. Finally, we apply our method for entanglement verification to existing experimental data, and use it to prove the existence of tripartite entanglement in a spin-squeezed atomic ensemble.