Bell correlation depth in many-body systems

F. Baccari, J. Tura, M. Fadel, A. Aloy, J.-D. Bancal, N. Sangouard, M. Lewenstein, A. Acín, and R. Augusiak

Phys. Rev. A 100, 022121 – Published 22 August 2019

Abstract

We address the question of assessing the number of particles sharing genuinely nonlocal correlations in a multipartite system. While the interest in multipartite nonlocality has grown in recent years, its existence in large quantum systems is difficult to confirm experimentally. This is mostly due to the inadequacy of standard multipartite Bell inequalities to many-body systems: Such inequalities usually rely on expectation values involving many parties, usually all, and require individual addressing of each party. In addition, known Bell inequalities for genuine nonlocality are composed of a number of expectation values that scales exponentially with the number of observers, which makes such inequalities impractical from the experimental point of view. In a recent work [Tura et al., Science 344, 1256 (2014)], it was shown that it is possible to detect nonlocality in multipartite systems using Bell inequalities with only two-body correlators. This opened the way for the detection of Bell correlations with trusted collective measurements through Bell correlation witnesses [Schmied et al., Science 352, 441 (2016)]. These witnesses were recently tested experimentally in many-body systems such as Bose-Einstein condensate or thermal ensembles, hence demonstrating the presence of Bell correlations with assumptions on the statistics. Here, we address the problem of detecting nonlocality depth, a notion that quantifies the number of particles sharing nonlocal correlation in a multipartite system. We introduce a general framework allowing us to derive Bell-like inequalities for nonlocality depth from symmetric two-body correlators. We characterize all such Bell-like inequalities for a finite number of parties and we show that they reveal Bell correlation depth $k \leq 6$ in arbitrarily large systems. We then show how Bell correlation depth can be estimated using quantities that are within reach in current experiments. On one hand, we use the standard multipartite Bell inequalities such the Mermin and Svetlichny ones to derive Bell correlations witnesses of any depth that involves only two collective measurements, one of which being the parity measurement. On the other hand, we show that our two-body Bell inequalities can be turned into witnesses of depth $k \leq 6$ that require measuring total spin components in certain directions. Interestingly, such a witness is violated by existing data from an ensemble of 480 atoms.

Received 22 March 2019

DOI:https://doi.org/10.1103/PhysRevA.100.022121

Physics Subject Headings (PhySH)

Nonlocality Quantum entanglement Quantum networks

Bose-Einstein condensates

Quantum Information, Science & TechnologyGeneral PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

F. Baccari¹, J. Tura^1,2, M. Fadel³, A. Aloy¹, J.-D. Bancal³, N. Sangouard³, M. Lewenstein^1,4, A. Acín^1,4, and R. Augusiak ⁵

¹Institut de Ciencies Fotoniques (ICFO), Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
²Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
³Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
⁴ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain
⁵Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland

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Vol. 100, Iss. 2 — August 2019

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Images

Figure 1
A possible $L_{k}$ partition corresponding to a vertex of $P_{N, k}^{2, S}$ , for $N = 22, k = 3$ , and $L = 12$ . Here, we have taken a $12_{3}$ partition consisting of five subsets of size 1, four subsets of size 2, and three subsets of size 3. Each of the five parties that are alone can choose one out of the $n_{1} = 4$ possible local deterministic strategies. Counting how many particles choose the $i th$ strategy, where $1 \leq i \leq n_{1}$ , determines $ξ_{1, i}$ . Each pair of particles highlighted in yellow can choose one out of the $n_{2}$ possible Popescu-Rohrlich (PR) boxes of ${N S}_{2}$ and each triplet of particles highlighted in gray can choose one out of the $n_{3}$ PR boxes highlighted in gray. The remaining coordinates of $\vec{ξ}$ are obtained analogously by counting. In Fig. 6, a detailed explanation is given, showing how to obtain the values (15) and (B9) from $\vec{ξ}$ .
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Figure 2
A cartoon picture illustrating the cases when projection and intersection of a polytope with a hyperplane are the same operation (left) and the generic case in which the intersection of a polytope with a hyperplane is strictly contained into its projection onto the same hyperplane (right).
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Figure 3
Plot of the quantum violation $(β_{Q} - β_{1}) / 2 N$ (black line) of the inequality (23) obtainable by following the procedure in Ref. [25]. The violation is compared with the—appropriately rescaled— $k$ -producible bounds $(β_{k} - β_{1}) / 2 N$ for the same inequality (colored lines) for values $k = 4, 5, 6$ (curves shown from top to bottom). Recall that the bounds for $k \leq 3$ coincide with the local one, and hence they are not shown in the plot.
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Figure 4
Quantification of the Bell correlation depth in a BEC with inequality (34) and connection to spin squeezing and entanglement. Black: the data reported in Ref. [23] expressed in terms of the Rabi contrast $C_{b}$ and the squeezed second moment $ζ_{a}^{2}$ , with $1 σ$ error bars. The number of particles is $N = 480$ . Blue shaded region: Bell correlations detected by violation of inequality (34) for $k = 1$ . Red shaded region: entanglement witnessed by spin squeezing [54, 56]. Red lines: limits on $ζ_{a}^{2}$ below which there is at least $(k + 1)$ -particle entanglement [55], increasing in powers of 2 up to $k = 256$ . Blue lines: limits on $ζ_{a}^{2}$ below which there are Bell correlations of depth at least $k + 1$ , for $k = 1, ..., 6$ .
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Figure 5
Example of projection of a triangle onto the $y$ axis. The vertices of the triangle are the set ${(0, 0), (2, 0), (1, 1)}$ , whose projection are the points ${(0, 0), (0, 1)}$ . Therefore, the desired projected polytope is the set $0 \leq y \leq 1$ .
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Figure 6
An example with $N = 22$ , and the 3-partition into $n_{1} = 5$ sets of size 1, $n_{2} = 4$ of size 2, and $n_{3} = 3$ of size 3. The first sum in Eq. (B9) corresponds to the value of $S_{x y}$ that comes from the two-body correlators $〈 M_{x}^{(i)} M_{y}^{(j)} 〉$ within each set (i.e., $i, j \in A_{l}$ for some $l$ ). For the 1-body boxes, these values are clearly zero, and for larger boxes, they correspond to the two-body marginals of the corresponding Popescu-Rohrlich box (PR box). Therefore, once symmetrized, the contribution of the box involving $p$ parties using the $i th$ strategy is $S_{x y} (p, i)$ . The second sum in Eq. (B9) counts those two-body correlators $〈 M_{x}^{(i)} M_{y}^{(j)} 〉$ in which $i \in A_{k}, j \in A_{l}, k \neq l$ , and $| A_{k} | = | A_{l} | = p$ . These correlations are represented by the blue, red, and yellow lines. Because they are correlations coming from different PR boxes, the locality assumption guarantees the factorization $〈 M_{x}^{(i)} M_{y}^{(j)} 〉 = 〈 M_{x}^{(i)} 〉 〈 M_{y}^{(j)} 〉$ , yielding the term $S_{x} (p, i) S_{y} (p, i)$ once symmetrized. The factor $ξ_{p, i} (ξ_{p, i} - 1)$ is given by the fact that $S_{x y}$ is defined as the sum for all $i \neq j$ , therefore containing repetitions. Finally, the last sum in Eq. (B9) is given by all two-body correlators $〈 M_{x}^{(i)} M_{y}^{(j)} 〉$ in which $i \in A_{k}, j \in A_{l}$ , and $| A_{k} | = p, | A_{l} | = q$ with $p \neq q$ , i.e., two-body correlations connecting PR boxes of different sizes. Here, the locality assumption also enables a factorization which amounts to $S_{x} (p, i) S_{y} (q, j)$ once symmetrized, weighted by the number of occurrences $ξ_{p, i} ξ_{q, j}$ . These correspond to the purple, green, and orange lines in the figure.
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