Bipartite nonlocality with a many-body system

We consider a bipartite scenario where two parties hold ensembles of $1/2$-spins which can only be measured collectively. We give numerical arguments supporting the conjecture that in this scenario no Bell inequality can be violated for arbitrary numbers of spins if only first order moment observables are available. We then give a recipe to achieve a significant Bell violation with a split many-body system when this restriction is lifted. This highlights the strong requirements needed to detect bipartite quantum correlations in many-body systems device-independently.


Introduction
In a Bell test, distinct parties initially share a resource such as a quantum state. They are then given measurement settings that they use to obtain measurement outcomes. The joint statistics of their outcomes, conditioned on the settings, can then be used to reveal a number of properties. For instance, it can be shown that certain statistics are not compatible with pre-established agreements, as highlighted by the violation of a Bell inequality [1]. Beyond its fundamental interest, the violation of a Bell inequality is a valuable resource e.g. to guarantee that the outcomes are random. Interestingly, the randomness guarantees are independent of the details of the actual implementation when they are derived from a Bell inequality violation. More broadly, there is an entire research field aiming to exploit Bell inequality violations for applied physics which is called device-independent quantum information processing. For both fundamental and applied reasons, it is thus interesting to understand which quantum states with which measurements are able to violate a Bell inequality.
The violation of a Bell inequality is possible with multi-partite states but most multi-partite Bell inequalities involve products of local observables of all parties. Exceptions have been found recently. In particular [2][3][4], showed that some multi-partite states are able to violate Bell inequalities using the product of two local observables. These inequalities paved the way for Bell tests with many-body systems where few-body correlations-correlations between a small number of local observables-are accessible only. So far, however, no violation of these inequalities could be observed experimentally since they require a single addressing of the constituent particles. This requirement have been lifted with the use of a Bell correlation witness (see below for details) which were used to demonstrate the presence of Bell correlations in many-body states [5][6][7]. This implies that if one could measure the constituent particle individually, then the observed correlations are strong enough to violate Bell inequalities presented in [2][3][4]. Such an individual addressing is unrealistic for a large number of particles. it is thus natural to ask whether a better solution exists to report on a Bell inequality violation with a many-body system.
Here, we make a step in this direction by considering the simple scenario in which hundreds of 1/2-spins are split among just two protagonists, Alice and Bob. We then ask whether a Bell violation could be observed when the parties are restricted to perform collective measurements on their 1/2-spin ensembles. This could be implemented with a large number of atoms constituting a Bose-Einstein condensate. A Bell inequality violation in this context would provide a strong demonstration that mesoscopic systems can behave differently from the predictions of classical physics. It would also show bi-partite quantum correlations between mesoscopic systems device-independently, i.e. independently of any assumption on the Hilbert space dimension and of the proper calibration of the measurement device.
As mentioned before, Bell tests involving collective measurements on many-body states could in principle be realized with a Bose-Einstein condensate. The basic idea is to use controlled interactions between the constituent bodies to create non-classical correlations between the internal states of these constituents [8,9], the later being essentially 1/2-spins. These spins could then be distributed [10,11] between Alice and Bob-n A spins for Alice and n B for Bob-before being measured, see figure 1. The experiment would then be repeated many times so that Alice and Bob can assess the expectation values of measurement results and demonstrate the Bell violation. In such a system, however, each protagonist can only measure his ensemble of particles collectively, that is, Alice can perform measurements of the form k is a vector having the 3 Pauli matrices as components and a  is a unit length vector with components α x , While entanglement witnesses have been intensively studied in this scenario with first order moments [12][13][14], few results are known with respect to Bell inequality violations. One noticeable exception is [15,16] where authors showed that bipartite correlations issued from first order collective measurements can be reproduced by a local model if the number of measurement settings is smaller or equal to n A for Alice and n B for Bob. In section 2, we give numerical arguments suggesting that such a bipartite Bell violation may not be possible when considering only first order moments, independently of the number of settings.
Beyond first order moments, it is worth noting that the use of collective measurements to violate a Bell inequality was also considered recently for a specific class of state with a strong tensor structure [17,18]. The state that we are interested in here are however very different: they do not admit any tensor structure but rather are typically symmetric under spin exchange. Significant Bell violations for large spin states using parity measurement have been found in [19,20] but using maximally entangled states, which are difficult to realise in practice. We show in section 3 that a Bell inequality can be violated using parity measurements on a state that can realistically be obtained in practice. We emphasize that a parity measurement requires the knowledge of a polynomial of high order moments of collective spin observables. Our work thus shows that performing collective spin observables in a split many-body system can lead to a Bell inequality violation provided that high order correlations can be assessed. Section 2 presents our attempts to find a bi-partite Bell inequality that can be violated with only first order moments of local collective spin components. Section 3 shows a concrete example where a Bell inequality violation is possible in a bi-partite scenario using collective spin projections provided that parity measurements can be performed. We conclude in section 4.

2.
Attempts to find a bi-partite Bell correlation witness involving only first order moments of local collective spin components The aim of this section is to study bi-partite Bell inequalities using only first order moments of collective spin components. Before we start, it is worth mentioning the result presented in [15,16] stating that no such inequality can be violated in our context when the number of settings is less than the number of local spins. Even if this puts stringent requirements on experiments on many-body systems, we want to know if this result holds without restriction on the number of settings. Furthermore, the issue on the number of settings can be circumvented in practice using Bell-correlation witnesses. We explain in section 2.1 the notion of Bell correlation witness after introducing the form of relevant Bell inequalities. Section 2.2 aims at computing the local bounds. In particular, we show that the local bounds are independent of the number of possible outcomes. The quantum bounds are the subject of section 2.3. Assuming collective measurements on 1/2 spins, the quantum bounds decreases while increasing the number of spins (outcomes). We use these two results in section 2.4 to give numerical arguments suggesting that it is not possible to violate a bi-partite Bell inequality using only first order moments of collective spin components.

Bell inequalities and Bell-correlation witnesses
Let us focus on a Bell scenario in which Alice and Bob each have a measurement box with m inputs A i and B j i, j ä [1, K, m] and N spins locally, i.e. N+1 possible outcomes --+ , 1 ..., .
We consider Bell inequalities of the form á ñ A i for example is the expectation value of outcomes for Alice's measurement corresponding to the input A i . w ij , v a i and v b j are weights, all taken in the interval {−1, 1} without lost of generality.
, , a b is the local bound, that is, the maximum value that the left hand side can take when considering bi-partite correlations steming from local models. This bound depends on the number of inputs and outcomes and on the weights, possibly in a nontrivial way, see below.
Note that if we assign to each input an observable  â A J i A i , one can associate to each instance of the inequality (2) a Bell operator so that a violation of " i j , leading to the well known Clauser, Horne, Shimony, and Holt (CHSH) inequality [21] for which the local bound is 2, that is, Assigning A 1 to σ x , A 2 to σ z , and B 1/2 to s s  ( ) 2 x z which correspond to the setting choice maximizing the CHSH value for the singlet, we get the Bell-correlation witness The latter can be evaluated with two collective measurements while the former requires the assessment of four correlators.
Recently, such a reduction in the number of measurements, applicable when assuming that the measurements are known and trusted, was used to transform a Bell inequality involving an unbounded number of settings into a witness with only two global measurements [3]. It was also used to show that the Svetlichny inequality [22], which is a Bell inequality for N parties requiring the measurement of 2 N correlators, reduces to a negativity condition on the mean value of an observable that can be evaluated with 2 measurement settings only [7].
There are thus at least two reasons for looking for Bell inequalities of the form (2) despite the result from [15,16] stating that no such inequality can be violated in our setting when the number of settings is less than the number of local spins, i.e.  m N . First, for every fixed value of N, there can exist a finite value of m>N potentially allowing for a Bell violation. Second, in a scenario in which one would be happy to trust the quantum description of the measurements, a Bell inequality with m>N, although involving a large number of settings could lead to a witness for bipartite Bell correlations that would require a maximum of three measurements per party. Such a witness would allow for the detection of a strong form of quantum correlations in many-body systems. Compared to previously known Bell correlation witnesses which only demonstrate the presence of some form of Bell correlations among a large number of spins, this witness would demonstrate Bell correlations between two well-identified spin ensembles: between Alice's set of spins and Bob's. We show in the next two subsections how to compute the local and quantum bounds respectively.

Local bound
In order to determine the local bound , , a b of the inequality (2), one has to consider all possible local deterministic strategies, that is, strategies assigning locally an outcome taken from -- 2 for each of the m settings. The local bound is simply the largest value that can be obtained from these deterministic strategies. On paper, it is sufficient to list all possible deterministic strategies to find the local bound. However, there are (N+1) 2m deterministic strategies, which makes the computation of the local bound complicated even for small N and m.
The number of relevant deterministic strategies is strongly reduced given the linearity of the inequality (2) with respect to the outcomes of Alice and Bob. To see this, let us consider an arbitrary deterministic strategy fixing the outcomes of Bob. The value of , , a b is obtained from a quantity of the form In the same way, we can demonstrate that for any strategy of Alice the maximum value of depending on the strategy of Alice. This implies that the optimal deterministic strategy is such that Remarkably, this observation applies to all Bell inequalities of the form (2): the maximum local value of such inequalities can be obtained by strategies involving only the two extremal outcomes, for all Conversely, it is possible to design an inequality of the form (2) whose local bound is achieved by only one such strategy. This implies that the extremal vertices of the local polytope within the space parametrized by á ñ á ñ A B , i j and á ñ A B i j are strategies assigning value  N 2 to all A i , B j . This polytope is thus isomorphic to the bipartite Bell scenario with m settings and 2 possible outcomes. The number of extremal strategies for this polytope is 2 2m , which is independent of the number of spins N.
In order to make the local bound independent of the number of spins we rescale the possible outcomes of A i and B j by N, defining a i =A i /N, b j =B j /N. This means that we now consider Bell inequalities of the form where this time the N+1 possible outcomes of Alice and Bob take value in --+ ¼ , , , The Bell operator corresponding to this inequality can be built in terms of the re-normalized spin operators with eigenvalues between - are the normalized spin projections: This operator is such that a violation of the inequality witnesses Bell correlations.
We thus focus now on Bell inequalities of the form (8). This form has the advantage that the local bound is independent of N. Once the local bound of such a Bell inequality is found, we want to show that it is a non-trivial Bell inequality by checking that it can be violated, that is, it admits a quantum value larger than its local bound.

Collective qubit bound
The maximal value of the inequality achievable by a quantum states of N plus N 1 2 -spins with local collective measurements  r N a b can be obtained by finding the maximum eigenvalue of the operator , , a b decreases with N. Given this and the fact that the local bound is independent of N, we focus on the case N=2. Finding no Bell violation for N=2 and an arbitrary number of settings is sufficient to show that there is no non-trivial Bell inequality for any N. We start by showing that indeed, the quantum bound  á ñ , , a b can be written in terms of the normalized spin projections as whereŴ is a 3×3 matrix having elements W xy given by to denote a subset of M<N spins and introduce the normalized spin observables over these spins This allows us to write our Bell operator for two sets Noticing that the spin projections for M spins satisfy å åå å Therefore, the maximum quantum value of the Bell operator for M<N spins per side bounds the value of the Bell operator with N spins: where to go from the second to the third line, we let the optimization over the state be independent for each term in the sum. This shows that the maximal value of  á ¢ ñ a b achievable with collective 1/2-spin measurements can only decrease with the number of spins N.

Numerical results
Let us start this subsection by a summary of the two previous subsections: (i) The local bound of an inequality of the form (8) is independent of the number of possible outcomes (ii) Assuming collective measurements on 1 2 -spins, the quantum bound decreases while increasing the number of spins (or outcomes). Together, these two statements imply that if a Bell inequality of the form (8) cannot be violated by performing collective measurements on an arbitrary state containing N particles on each side, then it is also impossible to violate it by performing collective measurements on a state with more than N particles on each side. Since the CHSH inequality is a non-trivial Bell inequality with N=1 spin locally, we focus on the case with N=2 spins locally. We know from [15,16] that in this case, one needs at least 3 measurements settings locally to circumvent known local models and thus possibly violate a Bell inequality of the form (8) with collective measurements.
In the case of 3 measurement settings, there is only one relevant Bell inequality with two outcomes [23]. Interpreting this inequality in terms of the normalized correlators of equation (8), we easily compute the maximum quantum value that a state of 4 particles (2 at each location) can achieve for this inequality by optimizing the maximal eigenvalue of the corresponding Bell operator as a function of the measurement settings. We find that this inequality is not violated (up to the accuracy of the computation) with collective spin measurement of 2 particles at each side. This implies that this inequality is likely not to admit a violation with collective spin measurement irrespectively of the number of spins per side.
We now consider the case of 4 measurement settings. In this case, the full polytope with binary outcomes has been recently shown to contain 175 different orbits up to relabellings of parties, inputs and outputs [24,25]. By gathering inequalities known in the litterature [21,23,[26][27][28][29][30][31][32][33][34], plus some other inequalities found by ourselves and by colleagues, we obtained a description of 175 such inequivalent families. We list them for the first time in the appendix. They provide a full description of our polytope. Focusing on these inequalities we computed the quantum bound as before with N=2 spins locally but with 4 measurements locally. We did not find any violation, which suggests that no inequality of the form (8) with m=4 can be violated by collective spin observables.
For the case of 5 settings, the local polytope is not known even for the simplest case of binary outcomes. We thus proceed differently. This time, we sample different inequalities of the form (8), that is, we choose w ¢ , ij ¢ v a i and ¢ v b j at random and computed both the local bound and the quantum bound. Note that this time we are not restricting ourselves to binary outcomes. We test 400 000 Bell inequalities with w ¢ 4 , We do not find any violation. Repeating the same procedure for 6 settings locally, we again do not find any non-trivial Bell inequality. 6 settings is the maximum we succeeded to do because it becomes increasingly expensive to find the maximum quantum value. Also the parameter space increases with the number of settings, so one would require more and more random trials to span the space of inequalities when the number of settings increases. Still, altogether this suggests that none of the inequalities of the form (8) can be violated by collective spin measurements when  N 2. We have presented numerical arguments suggesting that it is not possible to violate an inequality of the form (8) with collective spin measurements with 3, 4, 5 and 6 measurements settings whenever the number of spins locally is larger than two. In the next section, we show that the violation of a bipartite Bell inequality is possible when parity measurements are performed locally, for all spin number N.

The scenario
The scenario is similar to the one before. An ensemble of 1/2-spins is created in a quantum state before being shared between Alice and Bob. Each of them performs collective measurements of their spins â J l but in opposition to the scenario of the previous section, there is no limit on the order of moments of collective spin components that can be measured. We assume in particular that Alice and Bob can assess precisely the parity of the measurement outcome at each run. Concretely, we consider an ensemble of N 1/2-spins encoded in the internal degree of atoms, that is, two atomic states 1 and 2. These spins are located in Alice's location. We thus callâ i andˆ † a i with iä{1, 2} the bosonic operators associated to each spin states so that the collective spin projections can be written as We further consider that initially the spins point in the x direction and then undergoes one-axis twisting [35,36]. This results in a spin-squeezed state where ñ |0 is the vacuum state for all modes. The particles are then shared between Alice and Bob with a beam splitter type Hamiltonian, that is Hereb i andˆ † b i are bosonic operators for the spins located at Bob's location. In practice, each of these steps can be realized with a Bose-Einstein condensate where spin squeezing can be created using elastic collisions in state dependent potentials [8,9]. The spatial splitting can then be done by slowly raising a barrier in a stateindependent potential as in [10,11].

Probability distribution
When Alice and Bob measure â J A and b J B respectively, the probability with which they find the eigenvalues a k a and b k b is given by This probability can be efficiently calculated using equations (24) and (25).

Results
We consider the violation of In equation (5) which uses two settings and two outcomes per party. While many strategies can be used to bin the measurement results, we could only find a violation for the parity binning, which corresponds to the measurement of for Alice andq f for Bob. We present results obtained in this case below.
First, we fix the total number of spins N and we optimize the value of (5) over the measurement settings for various values of χt. The result is shown in figure 2 for N=6 and N=10. For low χt, the violation increases until a value which depends on the number of atoms and then goes down whereas in the extreme squeezing regime the violation can go higher. We further investigate the extreme squeezing case where χt=π/2. Remarkably the violation increases with the atom number and seems to saturate very close to the maximum value achievable by quantum states 2 2, see figure 3. This implies that it is possible to self test a singlet state and Pauli measurements with collective observables. Although this result is unexpected, it is extremely challenging if not completely out of reach experimentally as the maximally squeezed state corresponds to a GHZ state when χt=π/2.
We thus focus on the regime where χt is small, which is the most relevant regime in practice. We fix χt=0.006 corresponding to a Wineland squeezing parameter

Conclusion
We investigated the possibility of detecting bipartite nonlocality in many-body systems. We devoted a special attention to the experimental realization by considering realistic measurements. In particular, we focused on collective measurements only where spins are all measured in the same direction locally. We showed numerical results suggesting that no-Bell inequality with first order correlators can be violated whenever such measurements act of  N 2 spins. We then proved that the CHSH-Bell inequality can be violated with collective measurements as long as parity measurements can be performed. This suggests that parity measurements is a key ingredient-even maybe necessary-to reveal bipartite correlations in many-body systems with Bell inequalities.

Acknowledgments
This work was supported by the Swiss National Science Foundation (SNSF) through the Grant PP00P2-179109. We also acknowledge the Army Research Laboratory Center for Distributed Quantum Information via the project SciNet.  Appendix. Complete list of facets for the local polytope in the Bell scenario [( ) ( ) ] 2, 2, 2, 2 , 2, 2, 2, 2 The local polytope for two parties and four binary settings per party was recently solved in [24]. This work demonstrates that the polytope admits 175 classes of inequalities, up to permutations of parties, inputs and outputs, but it does not provide a description of these inequalities. Still, some of them can be found in the litterature. First, the positivity constraint, stating that probabilities are larger than 0, is known to be a (trivial) facet of every local polytope. Nontrivial Bell inequalities involving fewer than 4 settings for each party also give rise to lifted Bell inequalities in this scenario. This includes the CHSH inequality [21], I 3322 [23], as well as the three I 4322 1,2,3 inequalities [26]. But several inequalities using four measurements settings per party were also already known. Such is the case for A 5 , A 6 , A 7 , AS 1 , AS 2 , AII 1 , AII 2 and the I i 4422 inequalities with iä{1, K, 20} [26][27][28][29][30][31], as well as the J i 4422 inequalities with iä{1, K, 129} from [32] and the symmetric S i 242 inequalities with iä{1, K, 52} from [33]. Some of these inequalities are equivalent to each other, thus forming altogether 163 distinct families.
Noticing that randomly changing a few coefficients of a Bell inequality typically results in a non-facet defining Bell inequality with high rank, we used the algorithm given in section 2.3 of [33] to find tight facets below such inequalities for a number of random modifications of these 163 inequalities. This allowed us to discover a few additional inequalities, which were completed by a list of inequalities found independently by N. Brunner [34]. We refer to these additional 12 inequalities as N i 4422 with iä{1, K, 12}. Altogether, this provides a complete list of 175 families of tight Bell inequalities, which we present here.
A Bell inequality for m=4 binary settings is defined by 25 parameters. Following the main text we use the correlation picture with outcomes a i , b j ä{−1, 1}. A Bell inequality can then be written as or equivalently, in table format  b b b b a g g g g a g g g g a g g g g a g g g g In the main text we used ij and ℓ=δ. The coefficients of the 175 classes of inequalities are given in table A1 together with the name under which they have been referred to earlier in the litterature. The first 6 inequalities are liftings from simpler Bell scenarios, after which come 169 inequalities which truly involve all four settings of both parties.