Entanglement and extreme spin squeezing of unpolarized states

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 spin-squeezing 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.

With an interest towards fundamental questions in quantum physics, as well as applications, larger and larger entangled quantum systems have been realized with photons, trapped ions, and cold atoms [1][2][3][4][5][6][7][8][9][10][11].Entanglement is needed for certain quantum information processing tasks [12,13], and it is also necessary to reach the maximum sensitivity in a wide range of interferometric schemes in quantum metrology [14].Hence, the verification of the presence of entanglement is a crucial but exceedingly challenging task, especially in an ensemble of many, say 10 3 − 10 12 particles [5][6][7][8][9][10][11].Moreover, in such experiments it is not sufficient to claim that "the state is entangled", we need also to know how entangled the system is.Hence, quantifying entanglement in large ensembles has recently been at the center of attention.In several experiments the entanglement depth was determined, i.e., the minimal number of mutually entangled particles consistent with the measurement data, reaching to the thousands [7][8][9][10][11].
In the many-particle case, it is typically impossible to address the particles individually, and only collective quantities can be measured.In this context, one of the most successful approaches to detect entanglement is based on the criterion [15] ξ 2 s := N where N is the particle number, J l = N n=1 j (n) l for l = x, y, z are the collective spin components, and j (n) l are single particle spin components acting on the nth particle.The criterion (1) is best suited for states with a large collective spin in the (ŷ, ẑ)-plane and a small variance (∆J x ) 2 in the orthogonal direction.Every multiqubit state that violates Eq. ( 1) must be entangled [15] and has been also called spin squeezed in the context of metrology [16,17], due to the fact that the variance of a spin component is reduced below what can be achieved with fully polarized spin-coherent states.
As a generalization of Eq. ( 1), a criterion has also been derived by Sørensen and Mølmer [18], allowing to detect the depth of entanglement of spin-squeezed states by violating the condition based on the function where L l are the spin components of a single particle with spin J.As explained in Ref. [18], the functions F J with J = kj provide a boundary region for states with an entanglement depth of at most k in the ( J z , (∆J x ) 2 )-plane and thus every multi-spin-j state that violates Eq. ( 2) must have a depth of entanglement of (k + 1) or larger.Spin squeezing has been demonstrated in many experiments, from cold atoms [7,[19][20][21][22][23][24][25][26] to trapped ions [27], magnetic systems [28,29] and photons [30].
Recently, the concept of spin squeezing has also been extended to include states that are not fully polarized, such as singlet states, planar squeezed states and Dicke states [8,[31][32][33][34].The last in particular, which are another important class of multipartite entangled states, are attracting increasing attention and are being produced in experiments with photons [35,36] and Bose-Einstein condensates [8,24,37].Suitable criteria to detect the depth of entanglement of Dicke states have also been derived [8,38,39], but either they are limited to spin-1/2 particles or they are not optimal in the sense that they do not detect all states that could be detected based on the measured quantities.
In this paper we present a very general condition that outperforms all the entanglement criteria mentioned above, since (i) it detects multipartite rather than only bipartite entanglement, (ii) is applicable to spin-j systems, for any j, (iii) works both for spin-squeezed states and Dicke states, and (iv) is an optimal condition.Such a criterion can be applied immediately in experiments with Dicke states in spinor condenstates [40].
Summary of the main results.-Weshow that the condition holds for states with an entanglement depth at most k of an ensemble of N spin-j particles.We also analyze the performance of our condition compared to other criteria in the literature.
In general, the function F J (X) appearing on the righthand side of Eq. ( 4) has to be evaluated numerically.However due to its convexity properties we can bound it from below with the lowest nontrivial order of its Taylor expansion, yielding spin-squeezing parameters similar to the one defined in Eq. (1).While states saturating Eq. (4) determine a curve in the ( J 2 y + J 2 z , (∆J x ) 2 )-plane, these analytic conditions correspond to tangents to this curve.Hence, we will call them linear criteria in the paper.A family of such conditions for states with an entanglement depth k or smaller (such that kj is integer) is the inequality A similar condition can obtained from the Sørensen-Mølmer criteria (2) as A direct comparison between ξ 2 and ξ 2 SM shows that Eq. ( 5) is more suitable for detecting the depth of entanglement of unpolarized states, such as Dicke states, and also that, compared to Eq. ( 6), it takes advantage of the anti-squeezing on (∆J y ) 2 for fully polarized states.Note also the similarity of Eq. ( 6) and Eq.(1).All these criteria are also generalized to the fluctuating-N case, following Ref.[41].
Optimal nonlinear criteria.-Wedistinguish various levels of multipartite entanglement based on the following definitions.A quantum state is k-producible if it can be written as i of k l ≤ k particles, where p i are probabilities.Clearly, 1-producible states are separable states.A state that is not k-producible is called (k + 1)entangled.The entanglement depth is k +1 whenever the state is (k + 1)-producible but not k-producible [18,42].For clarity we introduce the notation G J : X → F J ( √ X).Observation 1.-The inequality in Eq. ( 4) holds for all k-producible states of N spin-j particles.The condition can be used if J 2 y + J 2 z ≥ N j(kj + 1), while otherwise (∆J x ) 2 = 0 can be achieved.Thus, every state of N spin-j particles that violates Eq. ( 4) must be (k + 1)entangled.
Proof.-The key argument of the proof is that for pure k-producible states of N spin-j particles holds.Then, for pure k-producible states (∆J x ) 2 ≥ N jF kj (RHS) ≥ N jF kj (LHS) follows from the properties of the F J (X), where we used the notation RHS and LHS for the left-hand side and right-hand side of Eq. ( 7), respectively.The first inequality originates from the result of Sørensen-Mølmer, Eq. ( 2), while the second follows from monotonicity of F J .Finally, the inequality can be extended to mixed k-producible states since the functions G J are convex and the argument is linear under mixing the state.This fact is discussed together with the proof of Eq. ( 7) and the tightness of Eq. ( 4), in the supplemental material.These criteria are especially suited to detect states for which J 2 y + J 2 z is large and (∆J x ) 2 is small.A paradigmatic example for this is the unpolarized Dicke state in the x-basis, ρ Dicke = |J = N j, m x = 0 J = N j, m x = 0| which is detected as N -entangled.
Numerical computation of G J .-The value of G J (X) can be obtained from the ground states of which are spin-squeezed states [18], and is feasible even for J of the order of thousands.When determining G J (X) for a given X, we need to carry out an optimization over λ.However, to study G J (X) for some interval of X, the explicit optimization can be avoided, which makes it possible to carry out calculations for very large systems.We just need to calculate the ground states |φ λ of Eq. ( 8) for a sufficiently wide interval of λ, and then the points of the curve are obtained as The result is shown in Fig. 1.Correspondingly, the boundary for k-producible states in the ( J 2 y + J 2 z , (∆J x ) 2 )-plane is parametrized by (cf.Fig. 2) We mention that for , the function on the right hand side of the criteria can be obtained analytically.Substituting F 1 (X) = G 1 (X 2 ) into Eq.( 4), we can obtain an analytic 2-producibility condition for qubits and an analytic separability condition for qutrits.For higher J, G J (X) is not known analytically.
Based on uncertainty relations of angular momentum operators a lower bound on G J (X) can be obtained as which is not tight for small J and small X, but becomes optimal for large J and X close to 1, see Ref. [18].Complementary to that approximation, in the following we derive a lower bound on G J (X) that is optimal for X ≈ 0 and improves GJ at small X by a factor of 2.
Linear analytic criteria.-Forinteger J, we can compute the first terms of the Taylor expansion of G J (X) about X = 0 and, using its convexity, obtain the bound G kj (X) ≥ (G kj (0) + XG ′ kj (0)).In other words, we can compute the tangent to the k-producibility boundaries, in the vicinity of their intersection point with the horizontal axis.This expansion can be done by employing the perturbation series for H λ in powers of the parameter λ ≪ 1, since X = 0 also corresponds to λ = 0.In particular we have that |φ 0 is the eigenstate of L x with eigenvalue zero and x /m 2 and |m x being the eigenstates of L x with eigenvalue m.Thus, the ground state of H λ is from which we obtain X(λ) ≈ λ 2 (J + 1) 2 and G J (λ) ≈  4).The dashed and dotted lines are Eq.( 15), i.e., the tangent to the curve, and the criterion ( 14) respectively.In the inset, curves are shown for k-producibility for N = 20 spin-1 particles, for k = 1, 5, 9, 13, 17.
Observation 2.-The criteria in Eqs.(5,6) hold for all k-producible states of N spin-j particles such that J = kj is an integer number.Every state of N spin-j particles that violates them must be (k + 1)-entangled.
Proof.-From Eq. ( 12) with J = kj we can bound from below the criteria (4) with Eq. ( 5) by substituting . Analogously, by rewriting Eq. ( 2) in terms of G kj and using the bound (12) with X = J z 2 /N 2 j 2 we obtain (6).See Fig. 2 for a plot of Eq. ( 5) as the tangent to the boundary of 20-producibility for N = 200 spin j = 1 2 particles in the ( J 2 y + J 2 z , (∆J x ) 2 )-plane.Comparison with similar criteria.-Dueto the monotonicity of F J , for comparing Eq. (2) and Eq. ( 4) it suffices to compare the arguments of the functions.
In practice, Eq. ( 13) holding is the most common situation, especially when noise is present that increases the variances on the left hand side.As the two extremal (and practically most important) examples we mention that it holds for states such that (∆J y ) 2 + (∆J z ) 2 ≫ N kj 2 (i.e., states close to Dicke states) and for states such that 2 ) (i.e., typical fully polarized states).Note also that from Eq. ( 13) it is clear that our inequality takes advantage of the anti-squeezing of the variance in the orthogonal direction for fully polarized spin squeezed states.For example, simple numerics show that for spin-squeezed states obtained as ground states of H µ = J 2 z − µJ x the condition ( 13) is fulfilled for N = 1000 spin- 1  2 particles for all µ, if white noise is added to the state with a small noise fraction p = 0.01.
Furthermore, we compare our criteria with another important set of conditions that are designed to detect the entanglement depth near unpolarized Dicke states.These are linear criteria derived by Duan in Ref. [38], stating that holds for all k-producible states of N spin-1 2 particles.Any state that violates Eq. ( 14) is detected as (k + 1)entangled.In this case, we can compare it with the linear criteria (5), specialized for qubit-systems, i.e., with j = 1 2 It is easy to see that a violation of Eq. ( 14) implies a violation of Eq. ( 15) and thus that our conditions detect strictly more states (see also Fig. 2).Finally, we note that Eq. ( 4) with j = 1/2 is similar to the criterion for spin-1/2 particles used in the experiment described in Ref. [8].A key difference is that in Eq. ( 4), in the denominator of the fraction, the term N (N − k)j 2 = N (N − k)/4 appears, while in the formula of Ref. [8] there is the term N 2 /4.The difference between the two criteria is the largest when we examine highly entangled Dicke states or spin-squeezed states, and in the argument of F kj (X) we have a value close to X = 1.
In the vicinity of this point, the derivative of F kj (X) is very large, hence some improvement in the argument of F kj (X) makes the bound on the right-hand side of Eq. (4) significantly higher.As a consequence, the criterion ( 4) is optimal for noisy Dicke states of many particles even in k ∼ N case, while the criterion of Ref. [8] is optimal only when k ≪ N, and it does not detect the Dicke state as N -entangled.
Extension to fluctuating number of particles.-Formacroscopic ensembles of particles, e.g., N ∼ 10 6 , the total number is not under perfect control and is impossible to collect statistical data for fixed N .This issue has been studied by Hyllus et al. [41], who suitably generalized the definition of entanglement depth and, by exploiting the concavity of variance, also all the spinsqueezing criteria to the case of fluctuating number of particles.Using similar methods, let us consider density matrices ρ = N Q N ρ N , where ρ N are the density matrices in some fixed-N subspace and Q N are probabilities.Also, let us consider the global collective spin operators Then, the same spin-squeezing criteria, e.g., Eq. ( 2), hold in the fluctuating N case simply under the substitution N → N , i.e., as [41] In our case, we can apply a similar reasoning.Observation 4.-All k-producible states in the Hilbert space H = H N must satisfy the following inequality where and W ≥ 0 is required.
Proof.-Given a state ρ Jensen's inequality and N = N Q N N .Note that the operator W defined in Eq. ( 18) is simply a direct sum of J 2 y,N + J 2 z,N over all fixed-N subspaces, normalized with a factor 1/(N j − kj).Thus, to apply our condition in experiments with fluctuating number of particles, one needs to measure the spin operators and the particle number jointly at each shot, and then average over an ensemble to compute W .
Conclusions.-Wederived a set of criteria to determine the depth of entanglement of spin-squeezed states and unpolarized Dicke states, extending and completing both the results of Refs.[8,18].These generalized spin-squeezing conditions are valid even for an ensemble of spin-j particles with j > 1  2 , which is very useful, since most experiments are done with particles with a higher spin, e.g., with spin-1 87 Rb atoms.Since theory is mostly available for the spin- 1  2 case, pseudo spin-1 2 particles are created artificially such that only two of the levels are populated.While the spin-squeezing approach to entanglement detection is already widely used in such systems [7,8,[19][20][21][22][23][24][25][26]33], our criteria make it possible to study spin-squeezing in fundamentally new experiments.A clear advantage of using the physical spin is that it is typically much easier to manipulate than the pseudo spin- 1  2 particles.In future, it would be interesting to clarify the relation between generalized spin squeezing and metrological usefulness [44][45][46][47][48][49], and also compare our results with the complete set of spin-squeezing criteria of Ref. [50], that contain one additional collective observable, related to single-spin average squeezing.

SUPPLEMENTAL MATERIAL
Proof. of Eq. ( 7)-To prove Eq. ( 7), let us consider the expression (∆J y ) 2 + (∆J z ) 2 on pure k-producible states ρ = i p i ρ , where N k stands for the total number of groups.We have, due to the additivity of the variance for tensor products (S1) where the superscript (l) indicates the lth group, that is composed of k l particles.Note that we also have x ) 2 ≥ 0 we can neglect it and obtain that can be bounded as due to the fact that max l k l = k and that the expression inside the round brackets in Eq. ( S3) is positive.Furthermore, using Jensen's inequality in the form for m = x, y, z we obtain Tightness of Eq. ( 7)-In deriving Eq. (S5) we neglected the contribution of l (j (l) x ) 2 := X which is a positive collective operator related to single particle spin squeezing.Also, for pure product states in general we have −X ≤ −(∆J x ) 2 with equality holding in the case J x = 0.For simplicity we omitted such X from Eq. (S5), that is thus not completely tight.However, it is still tight in the large N limit and in all practical cases, i.e., for experiments with Dicke states or fully polarized states for which (∆J x ) 2 = O(N ) and ).As a consequence, the criterion (4) is optimal in the large N limit.
Properties of F J .-The functions F J (X) can be obtained from the optimal states ρ for the problem defined in Eq. (3), i.e., the states that minimize (∆L x ) 2 for a given L z .These can be found as ground states of H = (L x − λ x ) 2 − λL z with the two parameters λ x and λ.For each given λ we can take the minimum ground state energy varying over λ x and for an integer spin J such minimum is attained for λ x = 0. Thus, for integer J, F J (X) is giving the minimal J 2 x for a given value of J z .Since the set of physical states is convex, the set of points in the ( J z , J 2 x )−space corresponding to physical states is also convex.Hence, F J (X) is also a convex function and in particular its derivative λ(X) is monotonously increasing with X.Note that in Ref. [18] a different proof was presented.In principle, the derivative F ′ J (X) can be computed by numerical derivation of F J (X).However, it is much simpler to obtain F ′ J (X) for some range of X by plotting ( 1J L z φ λ , λ) for some range of λ [18].In other words, for X = 1 J L z φ λ the derivative is F ′ J (X) = λ.To show that also G J (X) is convex we observe that ) is a monotonously increasing function of X.We evaluate numerically the derivative G ′ J (X) by plotting ( 1 J 2 L z 2 φ λ , J 2 Lz φ λ λ) for a wide range of λ, cf.Fig. S1, and see explicitly its monotonicity.More in general one can check whether or not F J (X 1 α ) is convex for any exponent α.It can then be observed numerically (not shown) that F J (X 1 α ) is not convex for any α > 2.