Entanglement and extreme spin squeezing of unpolarized states

Abstract

We present criteria to detect the depth of entanglement in macroscopic ensembles of spin-j particles using the variance and second moments of the collective spin components. The class of states detected goes beyond traditional spin-squeezed states by including Dicke states and other unpolarized states. The criteria derived are easy to evaluate numerically even for systems of very many particles and outperform past approaches, especially in practical situations where noise is present. We also derive analytic lower bounds based on the linearization of our criteria, which make it possible to define spinsqueezing parameters for Dicke states. In addition, we obtain spin squeezing parameters also from the condition derived in (Sørensen and Mølmer 2001 Phys. Rev. Lett. 86 4431). We also extend our results to systems with fluctuating number of particles. consider a subgroup of  k N particles and define its total spin asWe also need to define a function F J via a minimization over quantum states of such a group aswhere L l are the spin components of the group. In practice, the minimum will be the same if we carry out the minimization over states of a single particle with a spin J [19]. Then, for all pure states with an entanglement depth of at most kholds. It is easy to see that (4) is valid even for mixed states with an entanglement depth of at most k since the variance is concave in the state and F J (X) is convex 5 . Thus, every state that violates (4) must have a depth of entanglement of ( ) + k 1 or larger. It is important to stress that the criterion (4) provides a tight lower bound on ( ) DJ x 2 based on á ñ J .z Spin squeezing has been demonstrated in many experiments, from cold atoms [7, 20-26] to trapped ions [27], magnetic systems [28] and photons [29], and in many of these experiments even multipartite entanglement has been detected using the condition (4) [7,[23][24][25][26]29].Recently, the concept of spin squeezing has been extended to unpolarized states [30][31][32][33][34]. In particular, Dicke states are attracting increasing attention, since their multipartite entanglement is robust against particle loss, and they can be used for high precision quantum metrology [8]. Dicke states are produced in experiments with photons [35,36] and Bose-Einstein condensates [8,37,38]. Suitable criteria to detect the depth of entanglement of Dicke states have also been derived. However, either they are limited to spin-1/2 particles [8,39] or they do not give a tight lower bound on ( ) DJ x 2 based on the expectation value measured for the criterion, concretely,with F J (X) defined as in equation (3) and J = kj as in (2). Our approach is motivated by the fact that equation (4) fails to be a good criterion for mixed states with a low polarization  á ñ + á ñ J J N j y z 2 2 22 . Thus, we consider the second moments á + ñ J J y z 2 2 instead, which are still large for many useful unpolarized quantum states, such as Dicke states. Using the second moments is advantageous even for states with a large spin polarization since c...

Quantum metrology with nonclassical states of atomic ensembles

“…In particular, the condition ξ 2 D < 1 can be used to detect entanglement close to Dicke states (Tóth et al, 2007), see also Raghavan et al (2001). The detection of multiparticle entanglement close to Dicke states for spin-1/2 particles has been studied by Duan (2011) and Lücke et al (2014) and for spin-J particles with J > 1/2 by Vitagliano et al (2017). The inequalities (48b)-(48d) and the further inequality Ĵ 2 ≤ (N/2)(N/2 + 1), which is valid for all quantum states (not only for separable states), form a system of conditions that defines a polytope in the three dimensional space with coordinates Ĵ2 n1 , Ĵ2 n2 , and Ĵ2 n3 (Tóth et al, 2007(Tóth et al, , 2009.…”

Entanglement certification from theory to experiment

“…A similar simple structure remains even for device-independent certification of entanglement based on collective measurements [284]. Entanglement depth is typically used as a quantifier of entanglement in spin-squeezed states, which can also be witnessed with spin-squeezing parameters by making use of the Legendre transform method [107][108][109]. The general picture is that one can find a hierarchy of bounds on some collective quantities which depend on the entanglement depth, like (∆J z ) 2 ≥ N jF J Jy N j , where F J is a certain convex function which can be obtained through Legendre transforms.…”

Exploring the thermodynamics of spin-1 Bose gases with synthetic magnetization

“…Collisions between the different internal states of the atoms allow spin-changing collisions that have been studied in detail in [7][8][9][10]. These collisions can notably be employed to generate spin squeezing [11][12][13], that can be used to overcome the quantum shot noise limit [14][15][16][17][18]. Spinor dynamics was also studied in two dimensional systems [15], and in the presence of periodic potentials and across the superfluid to Mott insulator transition [19].…”