Many-Body Bell Correlations

• Many-body Bell correlations are a framework that detects nonlocal quantum effects via violations of multipartite Bell inequalities using low-order collective observables.
• This approach employs symmetric two-body Bell inequalities and collective spin measurements to certify nonlocality even under noisy, finite-statistics conditions.
• It enables quantification of Bell-correlation depth in systems like Bose–Einstein condensates, spin chains, and quantum processors, ensuring scalable quantum applications.

Many-body Bell correlations characterize the emergence and quantification of nonlocal quantum correlations across extensive multipartite systems. Unlike conventional entanglement, which is a necessary but not sufficient resource for Bell nonlocality, many-body Bell correlations are rigorously signaled by violations of multipartite Bell inequalities. In the context of large ensembles—such as atomic Bose–Einstein condensates, spin chains, and programmable quantum processors—recent theoretical and experimental developments have established a robust framework for certifying and quantifying such nonclassical correlations using accessible observables, often involving only collective or few-body measurements. This framework enables not only the detection of Bell nonlocality but also the determination of its "depth" (the number of parties genuinely sharing nonlocal correlations), under realistic conditions that include finite statistics and noise.

1. Symmetric Two-Body Bell Inequalities and Collective Witnesses

The cornerstone for detecting many-body Bell correlations in large systems is the construction of symmetric Bell inequalities based only on one- and two-body correlators. For $N$ qubits, each with $m$ possible dichotomic measurements $M_k^{(i)} = \pm 1$, the relevant symmetrized correlators are: $$S_k = \sum_{i=1}^N \langle M_k^{(i)} \rangle,\qquad S_{kl} = \sum_{i\neq j} \langle M_k^{(i)}M_l^{(j)} \rangle\,.$$ Bell functionals of the form

$$I_{N,m} = \sum_{k=0}^{m-1} \alpha_k S_k + \frac{1}{2}\sum_{k,l=0}^{m-1} S_{kl} \ge -\beta_c,$$

with antisymmetric $\alpha_k = m-2k-1$ and classical (local-realist) bound $\beta_c = \lfloor m^2N/2 \rfloor$, serve as the basis for such inequalities (Wagner et al., 2017, Tura et al., 2015). A critical feature is that these inequalities are tight (facet-defining) for the symmetric projected polytope and can certify macroscopic nonlocality even when only low-order (one- or two-body) moments are accessible.

Experimentally friendly collective witnesses are derived by identifying each local observable with a collective spin measurement along direction $\vec d_k$, defining

$$\hat S_k = \frac{1}{2}\sum_{i=1}^N \vec d_k \cdot \vec\sigma^{(i)},$$

and expressing the needed correlators in terms of first and second moments of collective spin observables: $$S_k = 2\langle \hat S_k \rangle,\qquad S_{kl} = 2\langle \hat S_k \hat S_l + \hat S_l \hat S_k \rangle - N(\vec d_k \cdot \vec d_l).$$ Witnesses such as

$$\mathcal W_m = C_b \sum_{k=0}^{m/2-1} \alpha_k \sin\vartheta_k - (1-\zeta_a^2) \left[ \sum_{k=0}^{m/2-1} \cos\vartheta_k \right]^2 + \frac{m^2}{4} \ge 0$$

involve only two collective observables—$C_b = 2\langle\hat S_{\vec b}\rangle/N$ (contrast) and $\zeta_a^2 = 4\langle \hat S_{\vec a}^2 \rangle / N$ (spin fluctuation)—and allow negative values to unambiguously certify Bell correlations. The witnesses are optimized over measurement axes to maximize effectiveness.

2. Detection in Spin-Squeezed States and Scaling Properties

Spin-squeezed states, widely generated in atomic and solid-state ensembles, are particularly susceptible to these witnesses. For the minimal case $m=2$, the boundary separating Bell-correlated from local-realist states is (Wagner et al., 2017, Schmied et al., 2016): $$\zeta_a^2 \ge Z_2(C_b) = \frac{1}{2}\left( 1 - \sqrt{1 - C_b^2} \right).$$ In the infinite-setting ($m \rightarrow \infty$) limit, the criterion tightens to

$$\zeta_a^2 < Z_{\infty}(C_b) = 1 - \frac{C_b}{\mathrm{artanh}(C_b)},$$

indicating that, in principle, arbitrarily small values of spin squeezing ($\zeta_a^2 < 1$) suffice if the contrast $C_b$ is close to unity. Critical points of Ising-type models and grounds states of one-axis-twisting Hamiltonians achieve these boundaries, making them optimal for demonstrating many-body nonlocality.

The scaling with system size is linear: For critical parameter regimes (e.g., ground states near quantum critical points or in models with infinite-range couplings), the violation $Q_\mu$ of suitable Bell correlators increases linearly with $N$, so that a macroscopic fraction of the system participates in Bell-nonlocal correlations (Hamza et al., 4 Mar 2024, Fadel et al., 2018, Tura et al., 2015).

3. Quantification: Depth of Bell Correlations

The "Bell-correlation depth" $k$ defines the minimal number of parties genuinely sharing nonlocal correlations. A state exhibits $k$-depth Bell correlations if it cannot be decomposed as a product or mixture over blocks smaller than $k$ in any (generally nonsignaling) framework. For two-body inequalities, each $k$ yields a specific classical threshold $\beta_c^k$; violation beyond this threshold certifies depth at least $k+1$ (Baccari et al., 2018): $$\mathcal{I} < -\beta_c^k \implies \text{Bell-correlation depth} \ge k+1.$$ Explicit analytical formulas provide these thresholds (e.g., for $k\leq 6$ and any $N$), and such depth witnesses have been achieved experimentally, with 6-body Bell correlation depth established in a 480-atom condensate.

4. Finite-Statistics, Noise Robustness, and Certification Requirements

Reliable detection of many-body Bell correlations in experiments necessitates statistical analysis against the statistics loophole. For any estimator $T$ of the collective witness based on $M$ experimental runs, the probability of incorrectly certifying nonlocality can be bounded using Bernstein’s inequality. The requisite number of runs is

$$M \ge \frac{-2\,t_l\,t_u - (2/3)(t_u-t_l)t_0}{t_0^2} \ln(1/\varepsilon),$$

where $t_0$ is the observed sample mean, $t_{l,u}$ are bounds on outcomes, and $\varepsilon$ is the targeted confidence level (Wagner et al., 2017). Notably, the number of repetitions needed to achieve a given confidence grows only linearly with $N$, in sharp contrast to the exponential scaling encountered in device-independent protocols without symmetry.

The presence of moderate technical noise or imperfections (detection efficiency, reduced contrast) is mitigated by the structure of the witness: boundary curves shift smoothly, and free parameters (e.g., measurement angles) can be optimized for robustness. The estimate's variance can be empirically determined, and no further loophole-closing assumptions are required beyond the already stated ones.

5. Experimental Platforms and Applications

Symmetric, collective measurement protocols are directly implementable in Bose–Einstein condensates, ultracold atomic ensembles, trapped ions, and solid-state spin ensembles (including NV centers and rare-earth doped crystals), provided collective spin or parity readout is available (Wagner et al., 2017, Schmied et al., 2016). Recent experiments have implemented such witnesses using two collective measurements in an atomic condensate to demonstrate Bell correlations with $N \sim 500$ atoms, observing violations by several standard deviations above shot noise.

The theoretical framework is also applicable to programmable quantum circuits and near-term quantum processors, where variational minimization of Bell Hamiltonians or energy witnesses allows direct certification and quantification of Bell-correlation depth across as many as 24 superconducting qubits, surpassing classical thresholds with overwhelming statistical significance (Wang et al., 25 Jun 2024). Related approaches include data-driven semidefinite programming, enabling device-independent certification based solely on first- and second-moment measurement statistics (Müller-Rigat et al., 2020, Fadel et al., 2017).

6. Comparison to Alternative and Limiting Regimes

While nonlocality in systems with exponential decay of correlations is generically suppressed except at criticality—where only short-range regions may show significant violation—fully connected (infinite-range) and near-critical spin models at quantum phase transitions exhibit robust, scalable Bell violations (Vieira et al., 2020, Piga et al., 2019). For systems with translational invariance and only finite-range interactions, a critical interaction range emerges for scalable Bell nonlocality: $r_c=4$ in 1D chains, above which the Bell-violation scales linearly with system size, and below which the violation decays exponentially with $N$ (Płodzień et al., 2023).

Analytical results using mappings to holomorphic oscillator modes, and effective Schrödinger-like equations for permutation-symmetric Hamiltonians, further reveal the universality of these phenomena. The critical temperature and system-size scaling of Bell correlation "survivability" can be directly linked to the spectral gap of the underlying Hamiltonian (Hamza et al., 4 Mar 2024, Fadel et al., 2018).

7. Outlook and Theoretical Extensions

Certifying many-body Bell correlations with collective observables bridges device-independent nonlocality tests with experimentally accessible quantum many-body measurements. The framework is extensible to higher spin ($j>1/2$) ensembles, systems with fluctuating particle number, and to the detection of dimension witnesses in SU($d$) systems via polynomially invariant Bell inequalities (Müller-Rigat et al., 18 Jun 2024). Future work includes exploring violations at quantum criticality in non-equilibrium dynamics, adapting protocol for multi-outcome or high-dimensional systems, and employing robust witnesses in quantum simulation, random number generation, and quantum metrology.

The systematic approach—via symmetric, robust, low-order collective witnesses, supported by rigorous statistical certification—establishes the theoretical and practical basis for observing, quantifying, and exploiting many-body Bell correlations in complex quantum systems under realistic conditions.