Spin squeezing and entanglement for an arbitrary spin

Abstract

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-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 [A. Sørensen et al., Nature 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.

Generalized spin-squeezing inequalities for particle number with quantum fluctuations

“…In the next example, we study numerically entanglement of N spin 1 state. We find that, when restricting to a subspace, our inequalities show a clear advantage over spin squeezing inequalities [16,17] for N spin 1 particles.…”

“…For this particular case, we show that our inequalities can be put in a coordinate system independent form. In section V, our inequalities are compared to the original ones [1,17] in two different cases. In the first example given in section V, we show that it is incorrect, in general, to replace N by its expectation value in spin squeezing inequalities, and that our inequalities should be used instead.…”

“…[17], by considering the situation of arbitrary particle number fluctuations, including quantum and/or classical ones. This generalization is important and necessary in many experimental situations even where the SSR applies.…”

“…If the global spin state is squeezed, that is, if the fluctuation of one of its component is sufficiently small compared to the expectation of the other components, then it can be shown that the N -spin systems is entangled [13,14]. * ibrahim.saideh@u-psud.fr Tóth and collaborators [1,[15][16][17] have generalized such an approach providing a set of inequalities that are fulfilled for all separable state of the N spins system and thus are able to detect entanglement when violated.…”

Effects of phase shift on bipartite and multipartite correlations of a spin chain under dephasing

“…Quantum correlations transfer can be accomplished using several quantum systems such as spin chains [23], optical systems [24,25], and trapped atoms [26,27]. Recently, the bipartite and multipartite correlations of spin chain have been studied by several authors [28][29][30][31][32]. The scaling of quantum discord in spin models was studied analytically and it was pointed out that at nite temperature the block scaling of quantum discord satis es an area law for any two-local Hamiltonian [29].…”