Inferring Nonlinear Many-Body Bell Inequalities From Average Two-Body Correlations: Systematic Approach for Arbitrary Spin-$j$ Ensembles

Violating Bell’s inequalities (BIs) allows one to certify the preparation of entangled states from minimal assumptions—in a device-independent manner. Finding BIs tailored to many-body correlations as prepared in present-day quantum computers and simulators is however a highly challenging endeavor. In this work, we focus on BIs violated by very coarse-grain features of the system: two-body correlations averaged over all permutations of the parties. For two-outcome measurements, specific BIs of this form have been theoretically and experimentally studied in the past, but it is practically impossible to explicitly test all such BIs. Data-driven methods—reconstructing a violated BI from the data themselves—have therefore been considered. Here, inspired by statistical physics, we develop a novel data-driven approach specifically tailored to such coarse-grain data. Our approach offers two main improvements over the existing literature: (1) it is directly designed for any number of outcomes and settings; (2) the obtained BIs are quadratic in the data, offering a fundamental scaling advantage for the precision required in experiments. This very flexible method, whose complexity does not scale with the system size, allows us to systematically improve over all previously known Bell inequalities robustly violated by ensembles of quantum spin $1/2$; and to discover novel families of Bell inequalities, tailored to spin-squeezed states and many-body spin singlets of arbitrary spin-$j$ ensembles.

Popular Summary

Quantum computers allow for solving problems intractable with classical devices, and are fundamentally boosted by nonlocal quantum correlations, such as entanglement. Hence, revealing the structure of entanglement within quantum computers is key to assess the quantum advantage they provide. This however represents one of the hardest problems offered by quantum-information science, for which no practical solution exists. While the relevant measurements might be already available to experimentalists, theoreticians are having a hard time figuring out how to exploit them to unveil entanglement. Here, we focus on a most coarse-grained version of this problem, and develop a highly versatile algorithm, which learns—from the data themselves—the best way to reveal entanglement: in the form of a Bell inequality (BI).

BIs represent a powerful concept in the quantum-information toolbox to certify quantum computers: if violated, they prove that entanglement is present even if one remains completely agnostic about the physical systems being measured. But in general, the number of potentially violated BIs is exponentially large in the system size: therefore a practical approach must learn the relevant BI from the experimental data. To make this problem fully tractable, we restrict our attention to BIs, which are invariant under permutations of the subsystems composing the device. By averaging over the potential fine-grain structure of entanglement, one might fear oversimplifying the problem, and losing the ability to certify entanglement altogether. In contrast, this allowed us to discover classes of BIs, valid for arbitrarily many subsystems, which are robustly violated by paradigmatic many-body entangled states: spin-squeezed and spin-singlet states of individual spin-$j$ components. While previous works focused on spin-1/2 degrees of freedom, our algorithm breaches the scalability wall towards a much wider exploration of BIs in many-body systems.

Figure 1

Illustration of an ideal multipartite Bell test for entanglement certification. Right: a composite quantum system prepared by the source is shared among $N$ spatially separated observers (on the sketch, $N=3$), each of which chooses a measurement settings $a\in \{0,1,\dots ,k-1\}$, obtaining an outcome $s\in \{-j,-j+1,\dots ,j\}$. In this work, we focus on spin measurements on quantum spin-$j$ particles, sketched as Stern-Gerlach magnets oriented along directions ${\overrightarrow{n}}_a$—but Bell tests are independent of these assumptions. This procedure is repeated several times in order to accumulate statistics, where at each round the measurement settings and the observed outcomes may vary. If the observed statistics exhibit Bell’s nonlocal correlations, one certifies that the multipartite state is quantum-mechanically entangled. Left: in a Bell test, each subsystem $i$ is treated as a black box, with no assumption about the Hilbert space. The actual observables ${\hat{s}}_a^{(i)}$ being measured are attributed arbitrary input labels $a$, and the measurement outcomes $s$ have no physical meaning. Entanglement is therefore certified in a device-independent manner.

Figure 2

The (nonlinear) Bell inequalities obtained with our method, which involve ${\tilde{C}}_{ab}$ [Eq. (8b)], are tighter than standard (linear) Bell inequalities, which involve $C_{ab}$ [Eq. (7b)]. Here, this is illustrated for $N=10$ for the Bell inequality of Eq. (19), which strengthens the previously known Eq. (18) [8, 9]. We slice the five-dimensional space $(M_0,M_1,C_{00},C_{01}=C_{10},C_{11})$ along a randomly chosen $2d$ plane [specifically, along the $(\overrightarrow{u},\overrightarrow{v})$ plane with $\overrightarrow{u}=(0.07679,0.24372,-0.34906,-0.75359,0.49494)$ and $\overrightarrow{v}=(0.29167,-0.90583,-0.20783,-0.21443,-0.07226)$]; $x$ and $y$ are the coordinates within this $2d$ plane. The convex black region ${\mathbb{P}}_L$ is the polytope of LV models, the dashed blue line is the linear Bell inequality of Eq. (18) [8, 9], and the red solid line is the corresponding nonlinear Bell inequality of Eq. (19), constructed from the ${\tilde{C}}_{ab}$ correlations, with the same coefficients.

Figure 3

Bell’s inequality violation for $j=1/2$ spin-squeezed states, as a function of the measurement angle $\theta$. Dashed red line: relative violation of the Bell inequality of Eq. (19), which involves $k=2$ measurement settings at angles $\pm \theta$. Dashed-dotted-dotted green line: relative violation of the Bell inequality of Eq. (21), which involves a third measurement along the $y$ axis (see the sketch on the bottom-left corner), with $a=1$ in Eq. (21). Solid orange line: same inequality, for the optimal value of the parameter $a$. The quantum state, chosen from Ref. [9], has a mean spin $2⟨{\hat{J}}_x⟩/N=m_x=0.98$, and transverse collective-spin fluctuations $4\mathrm{Var}({\hat{J}}_y)/N={\chi }^2=0.272$ (assuming $⟨{\hat{J}}_y⟩=0$). For all inequalities, the absolute value of the relative violation is equal to the amount of white noise tolerated by the data to observe a nonzero violation (see text).

Figure 4

Entanglement detection for $j=1/2$ spin-squeezed states. Below each line, the corresponding entanglement criterion is violated. Dashed-dotted black line: Wineland spin=squeezing criterion; solid orange line: violation of the Bell inequality Eq. (21) with the optimal parameter $a$; dashed-dotted-dotted green line: the same Bell inequality with $a=1$ [12]; dashed red line: violation of the Bell inequality Eq. (19) [9]. Black star: experimental data from Ref. [9], assuming $⟨{\hat{J}}_y⟩=0$. Notice that the previously known results were in fact involving $⟨{\hat{J}}_y^2⟩$. We prove in this work that they remain valid with $\mathrm{Var}({\hat{J}}_y)$ instead of $⟨{\hat{J}}_y^2⟩$, leading to systematically tighter criteria. In particular, the data of Ref. [9] (black star), using the actual value of $⟨{\hat{J}}_y⟩$ in the experiment, would be lower along the vertical axis, and similarly for the data of Ref. [10] (not shown).

Figure 5

Bell’s inequalities for many-body spin singlets. As input quantum data [Eq. (27a)] to our algorithm, we consider $k=3$ spin measurements in the $x$-$y$ plane, forming angles $t_1$, $0$, and $-t_2$ with the $x$ axis [inset of (a)]. We find violated Bell inequalities as in Eq. (28), with a conventional normalization $\sum _{ab}{A}_{ab}^2+\sum _a[h_a^{(2)}{]}^2=1$, and a classical bound given by Eq. (30). (a)–(d) Difference between the classical bound and the quantum value [Eq. (31)], divided by $N$. (a) $j=1/2$; (b) $j=1$; (c) $j=3/2$; (d) $j=2$. The integers $1$–$5$ label inequivalent families of Bell inequalities, which are found in the different regions of parameters $(t_1,t_2)$ (boundaries between these regions are indicatively emphasized as dotted white lines). In general, we find $2j$ Bell inequalities inequivalent under relabeling of the outcomes. Labels 1 and 2, respectively, correspond to the coefficients of Eqs. (38a) and (40a). (e) Along $t:=t_1=t_2$ [white dashed line on (a)–(d)], bound on $\mathrm{Var}({\hat{J}}_x)/N$ to observe Bell nonlocality assuming $\mathrm{SU}(2)$ invariance in this measurement setting [Eq. (37)]. The maximum at $t=\pi /3$ for half-integer spins corresponds to label $1$, and the witness condition is that of Eq. (39). The right-most local maximum at $t=\arccos [1/(4j)]$ corresponds to label 2, and the witness condition is that of Eq. (41).