Entanglement and extreme planar spin squeezing

TL;DR

This paper develops a new entanglement-depth criterion optimized for planar quantum squeezed states, improving detection of entanglement depth and monitoring its dynamics in experimental settings.

Contribution

It introduces a novel entanglement criterion tailored for PQS states, enhancing detection over previous methods and applied to real experimental data.

Findings

The new criterion detects higher entanglement depth in PQS states.

It outperforms the S{ exto}rensen-M{ exto}lmer criterion in certain regimes.

The method successfully monitors entanglement dynamics in a recent experiment.

Abstract

We introduce an entanglement-depth criterion optimized for planar quantum squeezed (PQS) states. It is connected with the sensitivity of such states for estimating an arbitrary, not necessarily small phase. We compare numerically our criterion with the well-known extreme spin squeezing condition of S{\o}rensen and M{\o}lmer [Phys. Rev. Lett. 86, 4431 (2001)] and show that our condition detects a higher depth of entanglement when both planar spin variances are squeezed below the standard quantum limit. We employ our theory to monitor the entanglement dynamics in a PQS state produced via quantum non-demolition (QND) measurements using data from a recent experiment [Phys. Rev. Lett. 118, 233603 (2017)].

Tables

Table 1. Table 1: Values of ζ J 2 subscript superscript 𝜁 2 𝐽 \zeta^{2}_{J} for 0 ≤ J ≤ 27 0 𝐽 27 0\leq J\leq 27 .

$J$	${\zeta }_J^2$	$J$	${\zeta }_J^2$	$J$	${\zeta }_J^2$
1	0.45	10	0.26067	19	0.21111
2	0.44906	11	0.25262	20	0.20758
3	0.38945	12	0.2455	21	0.20428
4	0.35321	13	0.23913	22	0.20118
5	0.32779	14	0.23338	23	0.19826
6	0.30852	15	0.22815	24	0.19551
7	0.29318	16	0.22336	25	0.1929
8	0.28054	17	0.21896	26	0.19043
9	0.26986	18	0.21489	27	0.18809

Equations

$${\xi }_{\parallel }^2:=\frac{(\Delta J_{\parallel }{)}^2}{|⟨J_{\parallel }⟩|},$$

$${\xi }_{\parallel }^2\ge {\zeta }_J^2,$$

$${\int }_0^{2\pi }\frac{\mathrm{d}\varphi }{2\pi }(\Delta \varphi {)}^2/(\Delta \varphi {)}_{\mathrm{SQL}}^2=\frac{1}{2}{\xi }_{\parallel }^2.$$

$$G_k^{(j)}(X):=\frac{1}{kj}\underset{\stackrel{\varphi \in ({\mathbb{C}}^d{)}^{\otimes k}}{\frac{1}{kj}⟨L_y{⟩}_{\varphi }=X}}{\min }\left[(\Delta L_y{)}_{\varphi }^2+(\Delta L_z{)}_{\varphi }^2\right],$$

$$(\Delta J_{\parallel }{)}^2\ge ⟨N⟩j{\mathcal{G}}_k^{(j)}\left(\frac{|⟨J_{\parallel }⟩|}{⟨N⟩j}\right),$$

$${\mathcal{G}}_k^{(j)}(X)\ge X{\zeta }_J^2,$$

$$\mathcal{L}[(\Delta L_{\parallel }{)}_{\varphi }^2/kj](T):=\underset{\varphi }{\inf }[\frac{1}{kj}(\Delta L_{\parallel }{)}_{\varphi }^2-⟨T{⟩}_{\varphi }],$$

$${\mathcal{G}}_k^{(j)}(X):=\underset{\lambda }{\sup }\{\lambda X-\mathcal{L}[(\Delta L_{\parallel }{)}_{\varphi }^2/kj](\lambda L_y/kj)\},$$

$$\mathcal{L}[(\Delta L_{\parallel }{)}_{\varphi }^2/kj](\lambda L_y/kj)=\frac{1}{kj}\underset{s_y,s_z}{\min }[\underset{\varphi }{\min }⟨H_{s_y,s_z,\lambda }{⟩}_{\varphi }],$$

$$⟨H_{s_y,s_z,\lambda }{⟩}_{\varphi }=\sum _{J=0}^{kj}{a}_J^2⟨(L_y^{(J)}-s_y{)}^2+(L_z^{(J)}-s_z{)}^2-\lambda L_y^{(J)}{⟩}_{{\psi }_J},$$

$$(\Delta J_z{)}^2\ge Nj{F}_J\left(\frac{⟨J_y⟩}{Nj}\right)$$

$$\theta(t)=g\Big{[}J_{z}(t_{\rm e})\cos\phi-J_{y}(t_{\rm e})\sin\phi\Big{]}e^{-t_{\rm r}/T_{2}}+\theta_{0},$$

Full text

*Introduction.—*Detecting entanglement in large quantum systems is a major goal in quantum information science and underpins the development of quantum technologies horodeckirev ; Guhne2009Entanglement . Attention has now shifted toward the practical use of entanglement as a resource: in particular, entanglement-enhanced sensing using ensembles of $10^{3}-10^{12}$ atomic spins has emerged as a major application tothapellaniz14 ; 2016arXiv160901609P . In this context, spin-squeezing inequalities can be used to quantify entanglement-enhanced sensitivity. Standard treatment studies spin-squeezed states (SSS), characterized by a large spin polarization in the $y$-direction and a small variance in the $z$-direction, via the parameter $\xi_{s}^{2}:=~{}\tfrac{N(\Delta J_{z})^{2}}{|\langle J_{y}\rangle|^{2}}$, where $J_{v}=\sum_{n=1}^{N}j_{v}^{(n)}$ for $v=x,y,z$ are the collective spin components, $j_{v}^{(n)}$ are single particle spin operators, $N$ is the total number of atoms. Let us assume that the mean collective spin points in the $y$-direction, while $J_{z}$ is the spin component in the orthogonal direction with the smallest variance. Then, states with $\xi_{s}^{2}<1$ provide quantum enhanced sensitivity for estimating phases $\phi\approx 0$ due to small rotations around $J_{x}$ Kitagawa1993Squeezed ; Wineland1994Squeezed . Such states have been produced using various platforms, including cold atoms Hald1999Spin ; Fernholz2008Spin ; Orzel2386 ; Riedel2010Atom-chip-based ; Esteve2008Squeezing ; SchleierSmithPRL2010 ; Leroux2010 ; Gross2010Nonlinear ; BohnetJ2014 ; SewellPRL2012 ; Cox2016 ; HostenN2016 , trapped ions Meyer2001 , magnetic systems Auccaise2015 and photons MitchellExtremespinsqueezing2014 .

The metrological sensitivity is strongly connected to entanglement. According to the original spin squeezing criterion, $\xi_{s}^{2}<1$ also implies entanglement for atoms with spin $j=1/2$ Sorensen2001Many-particle . This was extended to systems with a higher spin by Sørensen and Mølmer, who introduced a method to quantify entanglement by means of the so-called depth of entanglement, i.e., the number of particles in the largest separable subset Sorensen2001Entanglement . Several other highly entangled states have recently been found useful for quantum metrology, such as Dicke states. These states are unpolarized, have a large value of $\langle J_{x}^{2}+J_{y}^{2}\rangle$, and a small variance in the $z$-direction. Dicke states can be included in a class of generalized spin squeezed states, and their degree of entanglement can be quantified through collective spin variances. Spin squeezing inequalities have been developed to characterize entanglement in such states Lucke2014Detecting ; vitagliano16 , which have been produced in experiments with photons Wieczorek2009Experimental ; Prevedel2009Experimental and Bose-Einstein condensates Lucke2011Twin ; Hamley2012Spin-nematic ; Lucke2014Detecting ; Hoang2016Characterizing ; Luo620 .

Here, we focus on a different class of states, called planar quantum squeezed (PQS) states, studied theoretically in HePRA2011 ; He2012 ; PuentesNJP2013 , and observed in a recent experiment ColangeloNat2017 ; ColangeloPQS2017 . These states have reduced spin variances in two directions, i.e., $(\Delta J_{\parallel})^{2}:=(\Delta J_{y})^{2}+(\Delta J_{z})^{2}$ is small, and a large in-plane polarization, i.e., $\langle J_{y}\rangle\approx Nj.$ They provide quantum-enhanced sensitivity in estimating phases generated by rotations about the $\hat{x}$ axis, without the need of first localizing the phase around $\phi\simeq 0$ and are useful for tasks such as tracking a changing phase shift, or simultaneous estimation of phase and amplitude beyond classical limits ColangeloNat2017 ; ColangeloPQS2017 . The planar squeezing parameter

[TABLE]

where $|\langle J_{\parallel}\rangle|:=\sqrt{\langle J_{y}\rangle^{2}+\langle J_{z}\rangle^{2}}$ is the in-plane polarization, was introduced by He et al. HePRA2011 ; He2012 to quantify such enhanced sensitivity and detect entanglement. While planar squeezed states have intriguing properties, the relation between their metrological usefulness and their multiparticle entanglement has not been explored so far.

In this paper, we show how to detect multipartite entanglement that is also useful metrologically for the phase estimation task above. In particular, we introduce a method to detect the depth of entanglement based on the planar squeezing parameter $\xi_{\parallel}^{2}.$ We present the condition

[TABLE]

where $\zeta^{2}_{J}$ is the minimum value of the planar squeezing parameter over single particle states of spin $J.$ We prove that for all spin-$j$ ensembles that contain groups of at most $k$-entangled particles, called $k$-producible Guhne2005Multipartite ; Sorensen2001Entanglement , Eq. (2) holds with $J=kj.$ Thus, $\xi^{2}_{\parallel}<\zeta^{2}_{J}$, implies a depth of entanglement of at least $k+1=J/j+1.$ We can even estimate at least how many particles must be in fully entangled $(k+1)$-particle groups. We stress that our criterion is very simple to use. We need to calculate $\zeta^{2}_{J}$ only once for the relevant range of $J,$ then Eq. (2) can be applied for entanglement detection without any additional numerical optimization.

Finally, we examine the usefulness of our criterion. We compare it to the Sørensen-Mølmer criterion numerically, and find that it detects a higher entanglement depth for a large class of quantum states. We also test our theory using data from a recent experiment in which a PQS state was generated via semi-continuous quantum non-demolition (QND) measurement ColangeloNat2017 ; ColangeloPQS2017 .

*Link between our parameter and metrological sensitivity.—*We consider a protocol in which a collective spin state is rotated about the $J_{x}$ axis and accumulates a phase $\phi$ such that the operator $J_{z}$ evolves as $J_{z}^{\rm out}=J_{z}^{\rm in}\cos\phi-J_{y}^{\rm in}\sin\phi$. Afterwards, the phase is inferred from repeated measurements of $J_{z}^{\rm out}$, with a sensitivity given by the error-propagation formula $(\Delta\phi)^{2}=(\Delta J_{z}^{\rm out})^{2}/|\partial_{\phi}\langle J_{z}^{\rm out}\rangle|^{2}$. We consider as a reference an input state with an uncertainty at the standard quantum limit (SQL), $(\Delta J_{y}^{\rm in})_{\rm SQL}^{2}=(\Delta J_{z}^{\rm in})_{\rm SQL}^{2}=\tfrac{1}{2}|\langle J_{\parallel}^{\rm in}\rangle|$ PuentesNJP2013 ; ColangeloPQS2017 . A SQL-limited state cannot beat the shot-noise limit corresponding to separable states, since $(\Delta\phi)_{\rm SQL}^{2}=|\langle J_{\parallel}^{\rm in}\rangle|/(\langle J_{z}^{\rm in}\rangle^{2}\cos^{2}\phi+\langle J_{y}^{\rm in}\rangle^{2}\sin^{2}\phi)\geq 1/N$. Hence, we can normalize the sensitivity with respect to the SQL and obtain that $(\Delta\phi)^{2}/(\Delta\phi)_{\rm SQL}^{2}=\left[(\Delta J_{z}^{\rm in})^{2}\cos^{2}\phi+(\Delta J_{y}^{\rm in})^{2}\sin^{2}\phi\right]/|\langle J_{\parallel}^{\rm in}\rangle|$. By averaging over the phase we find

[TABLE]

Thus, the planar squeezing parameter quantifies the average sensitivity enhancement compared to the SQL over the interval $0\leq\phi\leq 2\pi$.

*Entanglement criterion for planar squeezing.—*Following an approach similar to past works Sorensen2001Entanglement ; vitagliano16 , we derive a tight criterion to detect the depth of entanglement based on the above planar squeezing parameter by computing the function

[TABLE]

where $d=2j+1$, $j$ is the single-particle spin quantum number, $L_{v}$ are $k$-particle operators constructed from spin-$j$ operators, i.e., $L_{v}=\sum_{n=1}^{k}j_{v}^{(n)},$ where $j_{v}^{(n)}$ are single-particle spin components. First, we derive a criterion that contains a tight lower bound on the planar spin variance valid for all states with a depth of entanglement smaller than $k$.

**Observation 1.—**Every $k$-producible state of a spin-$j$ particle system with average number of particles $\langle N\rangle$ must satisfy the tight inequality

[TABLE]

where $\mathcal{G}^{(j)}_{k}$ is defined as the convex hull of (4). Thus, every state that violates Eq. (5) must have a depth of entanglement of at least $k+1.$

*Proof.—*For pure $k$-producible states of a fixed number of particles $N$ we have $(\Delta J_{\parallel})_{N}^{2}=\sum_{n}\left[(\Delta L^{(n)}_{y})^{2}+(\Delta L^{(n)}_{z})^{2}\right]\geq\sum_{n}k_{n}j\mathcal{G}^{(j)}_{k_{n}}(\langle L_{y}^{(n)}\rangle/k_{n}j)$, where the $L_{v}^{(n)}$ are collective operators of $0\leq k_{n}\leq k$ particles. The second inequality follows directly from the definition of $\mathcal{G}^{(j)}_{k_{n}}$. Now, we have to take into account the fact that $\mathcal{G}^{(j)}_{k_{n}}$ are (i) convex and (ii) decreasing for increasing the index, i.e., $\mathcal{G}^{(j)}_{r}\leq\mathcal{G}^{(j)}_{s}$ for $r\geq s$ and that $k\geq k_{n}$ and $\sum_{n}k_{n}=N.$ Then, follows $\sum_{n}k_{n}j\mathcal{G}^{(j)}_{k_{n}}(\langle L_{y}^{(n)}\rangle/k_{n}j)\geq\sum_{n}k_{n}j\mathcal{G}^{(j)}_{k}(\langle L_{y}^{(n)}\rangle/k_{n}j)\geq Nj\mathcal{G}^{(j)}_{k}(\langle J_{y}\rangle/Nj),$ where the first inequality comes from property (ii) and the second from property (i) and Jensen’s inequality. Clearly, if $N$ is divisible by $k$ then the inequality (5) is tight by construction.

Let us consider now a state with a non-zero particle number variance given as $\varrho=\sum_{N}Q_{N}\varrho_{N}$, where $\varrho_{N}$ are states with a fixed particle number $N$ and $Q_{N}$ are the corresponding probabilities. From the properties above and from the concavity of the variance follows also that for such a state $(\Delta J_{\parallel})^{2}\geq\sum_{N}Q_{N}(\Delta J_{\parallel})_{N}^{2}\geq\sum_{N}Q_{N}Nj\mathcal{G}^{(j)}_{k}(\langle J_{y}\rangle/Nj)\geq\langle N\rangle j\mathcal{G}^{(j)}_{k}(\langle J_{y}\rangle/\langle N\rangle j)$ holds, $\langle N\rangle=\sum_{N}Q_{N}N$ being the average particle number. $\hfill\Box$

Starting from the function $G^{(j)}_{k}(X)$ given in Eq.(4) we can construct numerically the convex hull $\mathcal{G}_{k}^{(j)}$ that will have properties (i) and (ii) mentioned in the proof, as we now explain.

*Numerical computation of $\mathcal{G}_{k}^{(j)}$.—*In order to detect the depth of entanglement with our criterion we need to carry out the optimization in Eq. (4). For for $k=1$ and $j=1,$ straightforward algebra yields $G_{1}^{(1)}(X)=\tfrac{3}{2}-X^{2}-\tfrac{1}{2}\sqrt{1-X^{2}}$. Analytical expressions are very hard to obtain even for the next simplest cases. Numerically the optimization can be carried out by finding the ground states of $H_{\lambda}=L_{y}^{2}+L_{z}^{2}-\lambda L_{y}$, where $L_{v}$ are the $k$-particle operators defined before, which can be written in the usual block-diagonal form, where each block acts in some spin-$J$ subspace. Hence, the higher the $k$, the more are the spin-$J$ subspaces involved. One can show that, due to these, $\mathcal{G}_{k}^{(j)}$ cannot increase with $k$, which is just property (ii) needed in the proof of Observation 1 (see Supplemental Material Note1 for more details). When the optimization in Eq. (4) is restricted to the subspace with the maximal spin $J=kj,$ we call the resulting function $G^{\rm sy}_{J}(X)$, since it is computed by an optimization restricted to the symmetric subspace of $k$-partite states. In Fig. 2, we present a concrete example. The function $G_{4}^{(1)}(X)$ is plotted together with its convex hull $\mathcal{G}_{4}^{(1)}(X)$ and $G^{\rm sy}_{4}(X).$ We can see that a simple linear function corresponding to a straight line on the figure can be used as a lower bound to $\mathcal{G}_{4}^{(1)}(X).$ This lower bound works in general, as we show in what follows.

*Linear lower bound.—*As outlined above, the computation of $\mathcal{G}_{k}^{(j)}(X)$ still requires some numerics, which can be hard for high $k$ and $j$. Here, we will simplify further this task by finding a suitable lower bound that requires only the numerical computation of $G^{\rm sy}_{J}(X)$ with $J=kj$ and is thus easier than computing the full $\mathcal{G}_{k}^{(j)}(X)$.

**Observation 2.—**A convex lower bound to the curve $G_{k}^{(j)}(X)$ defined as in Eq. (4) is given by

[TABLE]

where $\zeta^{2}_{J}:=\min_{|\psi_{k}\rangle}[(\Delta L_{y})_{\psi_{k}}^{2}+(\Delta L_{z})_{\psi_{k}}^{2}]/\langle L_{y}\rangle_{\psi_{k}}$ is the minimum value of the planar squeezing parameter over single particle states $|\psi_{k}\rangle$ of spin $J=kj$. The proof is given in the Appendix.

With this method we need only to determine $\zeta^{2}_{J}$ for the relevant range of $J$, which can be written as $\zeta^{2}_{J}=\min_{X}[G^{\rm sy}_{J}(X)/X]$. Thus, as a simple algorithm one can: i) Find the ground states $|\phi_{\lambda}\rangle$ of $H_{\lambda}$ restricted to the symmetric subspace; ii) compute $(\Delta L_{\parallel})_{\phi_{\lambda}}^{2}$ and $\langle L_{y}\rangle_{\phi_{\lambda}}$; and finally take $\zeta^{2}_{J}=\min_{\phi_{\lambda}}(\Delta L_{\parallel})_{\phi_{\lambda}}^{2}/\langle L_{y}\rangle_{\phi_{\lambda}}$, which is feasible until very large $J,$ up to the thousands Note1 . As an example the values of $\zeta^{2}_{J}$ up to $J=27$ are given in Table 1, while the qualitative behavior can be observed in the inset of Fig. 2. Note that Eq. (6) is a tight approximation only for $k\geq 2$, independently on $j$. For $k=1$ the original criterion given in Eq. (5), has to be used instead.

From Observations 1 and 2, we immediately obtain Eq. (2), which connects the metrological performance of the PQS states to their entanglement depth. Next, we show that apart from proving that the entanglement depth is $k+1$, we can also obtain information about how many particles are in fully entangled groups of $(k+1).$ This provides a simple interpretation of the degree of the violation of Eq. (2).

**Observation 3.—**Let us assume that the total polarization is equally distributed over all particles. Then, there is at least a fraction $f_{k+1}=(1-\xi^{2}_{\parallel}/\zeta^{2}_{J})$ of particles in fully entangled groups of $(k+1)$ or more, with $k$ given by $J/j$. The proof is given in the Supplemental Material, where the case of varying particle numbers is included in the model Note1 . We discuss that without the assumption of equally split polarization, the above statement still holds for almost totally polarized states, i.e., with $\langle J_{y}\rangle\approx Nj$ and that similar ideas work also for the Sørensen-Mølmer criterion 111See Supplemental Material at [URL]..

*Practical use of the criterion.—*Thus, our criterion can be employed to detect the depth of entanglement in states for which two collective spin variances are known, as well as the total in-plane polarization. With the same input information, it would be possible to consider also the Sørensen-Mølmer extreme spin squeezing condition 222Note that the Sørensen-Mølmer condition requires as input just one variance, that must be the one orthogonal to the polarization.. Then, we can numerically compare the two criteria and study in which cases our criterion is more suitable to detect entanglement. To do this we parametrize the states with the ratio $\alpha=(\Delta J_{z})^{2}/(\Delta J_{y})^{2}$ between the two spin variances, and the total in-plane polarization $\beta=\langle J_{y}\rangle/N$. Then, we plot the lower bound on $(\Delta J_{z})^{2}$ for $k=5$ and $j=1$ for various values of $\alpha$ and $\beta$ and see for which regions of the $(\alpha,\beta)$ plane our criterion detects a higher depth. The result is shown in Fig. 3, where we can observe that our criterion detects a higher degree of entanglement in most of the plane, especially whenever the two variances become equal. Vice versa, totally polarized states that are spin squeezed only along the $z$-direction are detected with a higher depth by Sørensen-Mølmer’s criterion. All these statements are valid also for other $k$ and $j$ values. Thus, our criterion is especially tailored for detecting PQS states and distinguish those from traditional spin squeezed states, which are optimally detected by Sørensen-Mølmer’s criterion. Furthermore, the linearity of our criterion makes it directly connected with improved sensitivity in phase estimations: a value of $\xi^{2}_{\parallel}$ below the threshold given by the constant $\zeta^{2}_{J}$ with $J=kj$ implies that: (1) the state must be $(k+1)$-entangled and (2) its average sensitivity to rotations about the axis orthogonal to the plane of squeezing as compared to the SQL (given by the parameter $\xi^{2}_{\parallel}$ itself as in Eq. (3)) is better than that of any state with depth of entanglement $k$ or lower.

Next, we employ our criterion (2) to analyze entanglement generated in a recent experiment in which a PQS state was produced in an ensemble of $N=1.75\times 10^{6}$ cold 87Rb atoms via semi-continuous quantum non-demolition (QND) measurements ColangeloPQS2017 . In Fig. 4 we plot the observed planar squeezing parameter $\xi^{2}_{\parallel}$ as a function of the measurement strength, parametrized by number of photons $N_{\rm L}$ used in the QND measurement. As $N_{\rm L}$ increases, the input spin-coherent state evolves into a planar squeezed state, with squeezing observed between $N_{\rm L}\simeq 2\cdot 10^{8}$ and $N_{\rm L}\simeq 3\cdot 10^{8}$ photons, after which the spin variances increase due to noise and decoherence introduced by off-resonant scattering of probe photons. We also plot the corresponding fraction $f_{k+1}$ of atoms in fully entangled groups of $(k+1)$ or more, detected using our criterion. We observe the corresponding increase in entanglement depth with $N_{\rm L}$ up to the optimum of $N_{\rm L}~{}=~{}2.47\times 10^{8}$ photons, after which entanglement is gradually lost. At the optimum $N_{\rm L}$ we observe a spin coherence $\langle J_{\parallel}\rangle~{}=~{}0.83~{}N$, and a planar squeezing parameter $\xi_{\parallel}^{2}=0.32\pm 0.02$. For comparison, using the criterion developed by He et al. HePRA2011 ; He2012 , one would detect a fraction $0.39$ of atoms in entangled states, without any information about the depth of entanglement. The details of the experiment are given in the Supplemental Material Note1 .

*Conclusions.—*We have introduced a new criterion suitable to detect the depth of entanglement in planar squeezed states, and to distinguish them from traditional spin squeezed states, detectable with Sørensen-Mølmer criterion Sorensen2001Entanglement . Our criterion is simple to evaluate and directly connected with the sensitivity of PQS states for phase estimations that do not require any prior knowledge of the phase. By numerical comparison, we have also shown that our criterion represents an important alternative to Sørensen-Mølmer’s suitable to detect entanglement in PQS states. Finally, we tested our criterion with data from a recent experiment in which a PQS state was generated via semi-continuous QND measurement ColangeloPQS2017 .

This work was supported by the EU (ERC Starting Grant 258647/GEDENTQOPT, CHIST-ERA QUASAR, COST Action CA15220, ERC project AQUMET (grant agreement 280169), QUIC (grant agreement 641122) and ERIDIAN (Grant Agreement No. 713682)), the Austrian Science Fund (FWF) through the START project Y879-N27, the Spanish MINECO/FEDER, MINECO (projects No. FIS2012-36673-C03-03 and No. FIS2015-67161-P, MAQRO (Ref. FIS2015-68039-P), XPLICA (FIS2014-62181-EXP) and Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0522)), the Basque Government (Project No. IT4720-10), the National Research Fund of Hungary OTKA (Contract No. K83858), the CERCA Programme / Generalitat de Catalunya, Catalan 2014-SGR-1295, and by Fundació Privada CELLEX.

*Appendix. Proof of Observation 2.—*Let us consider a general pure $k$-particle state $\left|\phi\right>=\sum_{J}a_{J}\left|\psi_{J}\right>,$ where each $|\psi_{J}\rangle$ is a single spin-$J$ particle state. We assume that the mean planar spin points into the $y$-direction. We now need that the collective $k$-particle spin components $L_{v}$ can be written as the direct sum of operators $L_{v}^{(J)}$ acting on spin-$J$ particle spaces with $0\leq J\leq kj$ (or $1/2\leq J\leq kj$ for odd $k$ and half-integer $j$). Then, since the collective angular momentum operators do not couple the different spin-$J$ subspaces of the total $k$-particle space to each other, we have $(\Delta L_{\parallel})_{\phi}^{2}\geq\sum_{J>0}a_{J}^{2}\langle L_{y}^{(J)}\rangle_{\psi_{J}}\tfrac{(\Delta L_{\parallel}^{(J)})_{\psi_{J}}^{2}}{\langle L_{y}^{(J)}\rangle_{\psi_{J}}}.$ Hence, $(\Delta L_{\parallel})_{\phi}^{2}\geq\sum_{J>0}a_{J}^{2}\langle L_{y}^{(J)}\rangle_{\psi_{J}}\zeta^{2}_{J}$ follows. Finally, we obtain $\sum_{J>0}a_{J}^{2}\langle L_{y}^{(J)}\rangle_{\psi_{J}}\zeta^{2}_{J}\geq\zeta^{2}_{J_{\rm max}}\sum_{J>0}a_{J}^{2}\langle L_{y}^{(J)}\rangle_{\psi_{J}}=J_{\rm max}X\zeta^{2}_{J_{\rm max}}$, where $J_{\rm max}=kj$, since $X=\tfrac{1}{kj}\langle L_{y}\rangle_{\phi}=\tfrac{1}{kj}\sum_{J>0}a_{J}^{2}\langle L_{y}^{(J)}\rangle_{\psi_{J}}$ and $\zeta^{2}_{J}\geq\zeta^{2}_{J^{\prime}}$ for $J\leq J^{\prime}$. The last property can be observed numerically, cf. Fig. 2(inset). Due to the concavity of the variance, the statement follows for mixed states. $\hfill\Box$