arXiv:quant-ph/0504005 · Physical Review Letters

Spin Squeezing Inequalities and Entanglement of N Qubit States

Jarosław K. Korbicz, J. I. Cirac, Maciej Lewenstein

Published Sep 14, 2005• 197 citations

Abstract

We derive spin squeezing inequalities that generalize the concept of the spin squeezing parameter and provide necessary and sufficient conditions for genuine 2-, or 3-qubit entanglement for symmetric states, and sufficient condition for N-qubit states. Our inequalities have a clear physical interpretation as entanglement witnesses, can be easily measured, and are given by complex but elementary expressions.

DOI: 10.1103/physrevlett.95.120502

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arXiv:quant-ph/0504005 · Physical Review Letters

Spin Squeezing Inequalities and Entanglement of N Qubit States

Jarosław K. Korbicz, J. I. Cirac, Maciej Lewenstein

Published Sep 14, 2005• 197 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.95.120502

Related papers

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

Jarosław K. Korbicz, Otfried Gühne, Maciej Lewenstein +3 more

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.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.

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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 $\mathrm{\ensuremath{\Delta}}{J}_{x}\mathrm{\ensuremath{\Delta}}{J}_{y}\ensuremath{\ge}|\ensuremath{\langle}{J}_{z}\ensuremath{\rangle}|/2$. 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 $J$ , as in Greenberger--Horne--Zeilinger (GHZ) states and W states where $J=1/2$ . These inequalities are derived from the planar spin-uncertainty relation ${(\mathrm{\ensuremath{\Delta}}{J}_{x})}^{2}+{(\mathrm{\ensuremath{\Delta}}{J}_{y})}^{2}\ensuremath{\ge}{C}_{J}$ where ${C}_{J}$ 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.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 $\mathrm{\ensuremath{\Delta}}{J}_{x}\mathrm{\ensuremath{\Delta}}{J}_{y}\ensuremath{\ge}|\ensuremath{\langle}{J}_{z}\ensuremath{\rangle}|/2$. 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 $J$, as in Greenberger--Horne--Zeilinger (GHZ) states and W states where $J=1/2$. These inequalities are derived from the planar spin-uncertainty relation ${(\mathrm{\ensuremath{\Delta}}{J}_{x})}^{2}+{(\mathrm{\ensuremath{\Delta}}{J}_{y})}^{2}\ensuremath{\ge}{C}_{J}$ where ${C}_{J}$ 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.

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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 ${J}_{k}$ . They can be used for the experimental detection of entanglement in a system of spin- $\frac{1}{2}$ 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 ${J}_{k}$ in three orthogonal directions, or (ii) the variances of ${J}_{k}$ in three orthogonal directions, or (iii) the mean values of ${J}_{k}^{2}$ in three orthogonal directions, or (iv) the mean values and variances of ${J}_{k}$ 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.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 ${J}_{k}$. They can be used for the experimental detection of entanglement in a system of spin-$\frac{1}{2}$ 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 ${J}_{k}$ in three orthogonal directions, or (ii) the variances of ${J}_{k}$ in three orthogonal directions, or (iii) the mean values of ${J}_{k}^{2}$ in three orthogonal directions, or (iv) the mean values and variances of ${J}_{k}$ 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.

arXiv:0806.1048Physical Review AApr 27, 2009222 citations

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.

\frac{1}{2}

particles to entanglement conditions for spin-

j

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.

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 $\mathrm{\ensuremath{\Delta}}{J}_{x}\mathrm{\ensuremath{\Delta}}{J}_{y}\ensuremath{\ge}|\ensuremath{\langle}{J}_{z}\ensuremath{\rangle}|/2$. 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

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 ${(\mathrm{\ensuremath{\Delta}}{J}_{x})}^{2}+{(\mathrm{\ensuremath{\Delta}}{J}_{y})}^{2}\ensuremath{\ge}{C}_{J}$ where

{C}_{J}

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.

{J}_{k}

. They can be used for the experimental detection of entanglement in a system of spin-

\frac{1}{2}

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

{J}_{k}

in three orthogonal directions, or (ii) the variances of

{J}_{k}

in three orthogonal directions, or (iii) the mean values of

{J}_{k}^{2}

in three orthogonal directions, or (iv) the mean values and variances of

{J}_{k}

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.