Multipartite Bell Inequalities Research Articles

Non-adaptive measurement-based quantum computation on IBM Q

We test the quantumness of IBM’s quantum computer IBM Quantum System One in Ehningen, Germany. We generate generalised n-qubit GHZ states and measure Bell inequalities to investigate the n-party entanglement of the GHZ states. The implemented Bell inequalities are derived from non-adaptive measurement-based quantum computation (NMQC), a type of quantum computing that links the successful computation of a non-linear function to the violation of a multipartite Bell-inequality. The goal is to compute a multivariate Boolean function that clearly differentiates non-local correlations from local hidden variables (LHVs). Since it has been shown that LHVs can only compute linear functions, whereas quantum correlations are capable of outputting every possible Boolean function it thus serves as an indicator of multipartite entanglement. Here, we compute various non-linear functions with NMQC on IBM’s quantum computer IBM Quantum System One and thereby demonstrate that the presented method can be used to characterize quantum devices. We find a violation for a maximum of seven qubits and compare our results to an existing implementation of NMQC using photons.

An elegant scheme of self-testing for multipartite Bell inequalities

Self-testing is the most accurate form of certification of quantum devices. While self-testing in bipartite Bell scenarios has been thoroughly studied, self-testing in the more complex multipartite Bell scenarios remains largely unexplored. We present a simple and broadly applicable self-testing scheme for N-partite correlation Bell inequalities with two binary outcome observables per party. To showcase the versatility of our proof technique, we obtain self-testing statements for the MABK and WWWŻB family of linear Bell inequalities and Uffink’s family of quadratic Bell inequalities. In particular, we show that the N-partite MABK and Uffink’s quadratic Bell inequalities self-test the GHZ state and anti-commuting observables for each party. While the former uniquely specifies the state, the latter allows for an arbitrary relative phase. To demonstrate the operational relevance of the relative phase, we introduce Uffink’s complex-valued N partite Bell expression, whose extremal values self-test the GHZ states and uniquely specify the relative phase.

Boosting device-independent cryptography with tripartite nonlocality

Device-independent (DI) protocols, such as DI conference key agreement (DICKA) and DI randomness expansion (DIRE), certify private randomness by observing nonlocal correlations when two or more parties test a Bell inequality. While most DI protocols are restricted to bipartite Bell tests, harnessing multipartite nonlocal correlations may lead to better performance. Here, we consider tripartite DICKA and DIRE protocols based on testing multipartite Bell inequalities, specifically: the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and the Holz and the Parity-CHSH inequalities introduced in the context of DICKA protocols. We evaluate the asymptotic performance of the DICKA (DIRE) protocols in terms of their conference key rate (net randomness generation rate), by deriving lower bounds on the conditional von Neumann entropy of one party's outcome and two parties' outcomes. For the Holz inequality, we prove a tight analytical lower bound on the one-outcome entropy and conjecture a tight lower bound on the two-outcome entropy. We additionally re-derive the analytical one-outcome entropy bound for the MABK inequality with a much simpler method and obtain a numerical lower bound on the two-outcome entropy for the Parity-CHSH inequality. Our simulations show that DICKA and DIRE protocols employing tripartite Bell inequalities can significantly outperform their bipartite counterparts. Moreover, we establish that genuine multipartite entanglement is not a precondition for multipartite DIRE while its necessity for DICKA remains an open question.

Persistent nonlocality in an ultracold-atom environment

We investigate nonlocal quantum correlations arising between multiple two-level impurity atoms coupled to an ultracold bosonic gas. We find that the environment-induced dynamics of the impurity subsystem can generate nonlocal states that are robust against noise and violate a multipartite Bell inequality when projective spin measurements are made. Genuine multipartite nonlocality is also observed in a system of three impurities. We show that non-Markovian effects, and the persistence of coherences in the impurity subsystem, are crucial for preventing complete loss of nonlocality and allow for nonlocal correlations to be generated and maintained for extended periods of time.

Constructing multipartite Bell inequalities from stabilizers

Bell inequality with a self-testing property has played an important role in quantum information processing with both fundamental and practical significance. However, it is generally challenging to find Bell inequalities used for self-testing multipartite states, and actually, there are not many known candidates. In this work we propose a systematic framework to construct Bell inequalities from stabilizers which are maximally violated by general stabilizer states, with two observables for each local party. We show that the constructed Bell inequalities can self-test any stabilizer state if and only if these stabilizers can uniquely determine the state in a device-dependent manner. This bridges the gap between device-independent and device-dependent verification methods. Our framework can provide plenty of Bell inequalities for self-testing $N$-party stabilizer states. Among them, we give two families of Bell inequalities with different advantages: (1) a family of Bell inequalities with a constant ratio of quantum and classical bounds using $2N$ correlations, and (2) single pair inequalities improving on all previous robustness self-testing bounds using $N+1$ correlations, which are both efficient and robust for realizations in multipartite systems.

Certification of Genuine Multipartite Entanglement with General and Robust Device-Independent Witnesses.

Genuine multipartite entanglement represents the strongest type of entanglement, which is an essential resource for quantum information processing. Standard methods to detect genuine multipartite entanglement, e.g., entanglement witnesses, state tomography, or quantum state verification, require full knowledge of the Hilbert space dimension and precise calibration of measurement devices, which are usually difficult to acquire in an experiment. The most radical way to overcome these problems is to detect entanglement solely based on the Bell-like correlations of measurement outcomes collected in the experiment, namely, device independently. However, it is difficult to certify genuine entanglement of practical multipartite states in this way, and even more difficult to quantify it, due to the difficulty in identifying optimal multipartite Bell inequalities and protocols tolerant to state impurity. In this Letter, we explore a general and robust device-independent method that can be applied to various realistic multipartite quantum states in arbitrary finite dimension, while merely relying on bipartite Bell inequalities. Our method allows us both to certify the presence of genuine multipartite entanglement and to quantify it. Several important classes of entangled states are tested with this method, leading to the detection of genuinely entangled states. We also certify genuine multipartite entanglement in weakly entangled Greenberger-Horne-Zeilinger states, showing that the method applies equally well to less standard states.

Robust violation of a multipartite Bell inequality from the perspective of a single-system game

Recently, Fan et al. [Mod. Phys. Lett. A 36, 2150223 (2021)] presented a generalized Clauser–Horne–Shimony–Holt (CHSH) inequality, to identify [Formula: see text]-qubit Greenberger–Horne–Zeilinger states. They showed an interesting phenomenon that the maximal violation of the generalized CHSH inequality is robust under some specific noises. In this work, we map the inequality to the CHSH game, and consequently to the CHSH* game in a single-qubit system. This mapping provides an explanation for the robust violations in [Formula: see text]-qubit systems. Namely, the robust violations, resulting from the degeneracy of the generalized CHSH operators correspond to the symmetry of the maximally entangled two-qubit states and the identity transformation in the single-qubit game. This explanation enables us to exactly demonstrate that the degeneracy is [Formula: see text].

Quantifying multipartite quantum entanglement in a semi-device-independent manner

We propose two semi-device-independent approaches that are able to quantify unknown multipartite quantum entanglement experimentally, where the only information that has to be known beforehand is quantum dimension, and the concept that plays a key role is nondegenerate Bell inequalities. Specifically, using the nondegeneracy of multipartite Bell inequalities, we obtain useful information on the purity of target quantum state. Combined with an estimate of the maximal overlap between the target state and pure product states, and a continuous property of the geometric measure of entanglement, we shall prove the information on purity allows us to give a lower bound for this entanglement measure. In addition, we show that a different combination of the above results also converts to a lower bound for the relative entropy of entanglement. As a demonstration, we apply our approach on 5-partite qubit systems with the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and show that useful lower bounds for the geometric measure of entanglement can be obtained if the Bell expression value is larger than 3.60, and those for the relative entropy of entanglement can be given if the Bell expression value is larger than 3.80, where the Tsirelson bound is 4.

On safe post-selection for Bell tests with ideal detectors: Causal diagram approach

Reasoning about Bell nonlocality from the correlations observed in post-selected data is always a matter of concern. This is because conditioning on the outcomes is a source of non-causal correlations, known as aselection bias, rising doubts whether the conclusion concerns the actual causal process or maybe it is just an effect of processing the data. Yet, even in the idealised case without detection inefficiencies, post-selection is an integral part of experimental designs, not least because it is a part of the entanglement generation process itself. In this paper we discuss a broad class of scenarios with post-selection on multiple spatially distributed outcomes. A simple criterion is worked out, called theall-but-oneprinciple, showing when the conclusions about nonlocality from breaking Bell inequalities with post-selected data remain in force. Generality of this result, attained by adopting the high-level diagrammatic tools of causal inference, provides safe grounds for systematic reasoning based on the standard form of multipartite Bell inequalities in a wide array of entanglement generation schemes, without worrying about the dangers of selection bias. In particular, it can be applied to post-selection defined by single-particle events in each detection chanel when the number of particles in the system is conserved.

Polytope structures for Greenberger–Horne–Zeilinger diagonal states *

We explore the polytope structures for genuine entanglement, biseparability, full biseparability and Bell inequality of multi-qubit GHZ diagonal states. We first show that biseparable GHZ diagonal states make hypersimplices inside the simplices consisting of all GHZ diagonal states. Next, we consider full biseparability which is equivalent to positive partial transpose for GHZ diagonal states, and show that they make the convex hulls of simplices and cubes. We also visualize which part of the simplex violates multipartite Bell inequality. Finally, we compute precise volumes for genuine entanglement, biseparability, full biseparability and states violating Bell inequality among all GHZ diagonal states.

Entropy Bounds for Multiparty Device-Independent Cryptography

Multiparty quantum cryptography based on distributed entanglement will find its natural application in the upcoming quantum networks. The security of many multipartite device-independent (DI) protocols, such as DI conference key agreement, relies on bounding the von Neumann entropy of the parties' outcomes conditioned on the eavesdropper's information, given the violation of a multipartite Bell inequality. We consider three parties testing the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of their outcomes by bounding the conditional entropy of a single party's outcome and the joint conditional entropy of two parties' outcomes. From the former bound, we show that genuine multipartite entanglement is necessary to certify the privacy of a party's outcome, while the latter significantly improve previous results. We obtain the entropy bounds thanks to two general results of independent interest. The first one drastically simplifies the quantum setup of an $N$-partite Bell scenario. The second one provides an upper bound on the violation of the MABK inequality by an arbitrary $N$-qubit state, as a function of the state's parameters.

Optimizing dichotomic local phase measurement settings for multipartite quantum systems

We investigate optimization of dichotomic local phase measurement settings for multipartite quantum systems. We first describe a method of constructing a single multipartite Bell inequality by treating the correlation function as an n-index tensor. Under a set of local phase measurement settings, we calculate the quantum prediction of the generalized Greenberger–Horne–Zeilinger state and show the exact relationship between quantum prediction and the phase factors. We show that there exists a set of optimal measurement settings relevant to maximal quantum prediction, which can contribute to testing multipartite Bell inequality in a straightforward manner. Also, we provide quantum simulation and numerical analysis with these optimal measurement settings.

Genuine multipartite Bell inequality for device-independent conference key agreement

In this work, we present a new class of genuine multipartite Bell inequalities, that is particularly designed for multipartite device-independent (DI) quantum key distribution (QKD), also called DI conference key agreement. We prove the classical bounds of this inequality, discuss how to maximally violate it and show its usefulness by calculating achievable conference key rates via the violation of this Bell inequality. To this end, semidefinite programming techniques based on [Nat. Commun. 2, 238 (2011)] are employed and extended to the multipartite scenario. Our Bell inequality represents a nontrivial multipartite generalization of the Clauser-Horne-Shimony-Holt inequality and is motivated by the extension of the bipartite Bell state to the n-partite Greenberger-Horne-Zeilinger state. For DIQKD, we suggest an honest implementation for any number of parties and study the effect of noise on achievable asymptotic conference key rates.

Strength and typicality of nonlocality in multisetting and multipartite Bell scenarios

In this work we investigate the probability of violation of local realism under random measurements in parallel with the strength of these violations as described by resistance to white noise admixture. We address multisetting Bell scenarios involving up to 7 qubits. As a result, in the first part of this manuscript we report statistical distributions of a quantity reciprocal to the critical visibility for various multipartite quantum states subjected to random measurements. The statistical relevance of different classes of multipartite tight Bell inequalities violated with random measurements is investigated. We also introduce the concept of typicality of quantum correlations for pure states as the probability to generate a nonlocal behaviour with both random state and measurement. Although this typicality is slightly above 5.3\% for the CHSH scenario, for a modest increase in the number of involved qubits it quickly surpasses 99.99\%.

Bell correlation depth in many-body systems

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 and require an individual addressing of each party. In a recent work [J. Tura et al. Science 344, 6189 (2014)] some of us proposed simpler Bell inequalities overcoming such difficulties, opening the way for the detection of Bell correlations with trusted collective measurements through Bell correlation witnesses [R. Schmied et al. Science 352, 441 (2016)], hence demonstrating the presence of Bell correlations with assumptions on the statistics. Here, we address the question of assessing the number of particles sharing genuinely nonlocal correlations in a multipartite system. This endeavour is a priori challenging, as known Bell inequalities for genuine nonlocality suffer from the above shortcomings, plus a number of measurement settings scaling exponentially with the system size. We first show that most of these constraints drop once the witnesses corresponding to these inequalities are expressed: in systems where multipartite expectation values can be evaluated, these witnesses can reveal genuine nonlocality for an arbitrary number of particles with just two collective measurements. We then introduce a general framework focused on two-body Bell-like inequalities. We show that they also provide information about the number of particles that are genuinely nonlocal. Then, we characterize all such inequalities for a finite system size. We provide witnesses of Bell correlation depth $k\leq6$ for any number of parties, within experimental reach. A violation for depth $6$ is achieved with existing data from an ensemble of 480 atoms.

Device-independent quantum secret sharing in arbitrary even dimensions

We present a device independent quantum secret sharing scheme in arbitrary even dimension. We propose a $d$-dimensional $N$-partite linear game, utilizing a generic multipartite higher dimensional Bell inequality, a generalization of Mermin's inequality in the higher dimension. Probability to win this linear game defines the device independence test of the proposed scheme. The security is proved under causal independence of measurement devices and it is based on the polygamy property of entanglement. By defining $\epsilon_{cor}$-correctness and $\epsilon^c$-completeness for a quantum secret sharing scheme, we have also shown that the proposed scheme is $\epsilon_{cor}$-correct and $\epsilon^c$-complete.

Periodic Fourier representation of Boolean functions

In this work, we consider a new type of Fourier-like representation of Boolean function f\colon\{+1,-1\}^n\to\{+1,-1\}: f(x) = \cos(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i). This representation, which we call the periodic Fourier representation, of Boolean function is closely related to a certain type of multipartite Bell inequalities and non-adaptive measurement-based quantum computation with linear side-processing NMQCp. The minimum number of non-zero coefficients in the above representation, which we call the periodic Fourier sparsity, is equal to the required number of qubits for the exact computation of f by \NMQCp. Periodic Fourier representations are not unique, and can be directly obtained both from the Fourier representation and the F_2-polynomial representation. In this work, we first show that Boolean functions related to ZZZZ-polynomial have small periodic Fourier sparsities. Second, we show that the periodic Fourier sparsity is at least 2^{\deg_{\mathbb{F}_2}(f)}-1, which means that NMQCp efficiently computes a Boolean function $f$ if and only if F_2-degree of f is small. Furthermore, we show that any symmetric Boolean function, e.g., AND_n, Mod^3_n, Maj_n, etc, can be exactly computed by depth-2 NMQCp using a polynomial number of qubits, that implies exponential gaps between NMQCp and depth-2 NMQCp.

Bell correlations at finite temperature

We show that spin systems with infinite-range interactions can violate at thermal equilibrium a multipartite Bell inequality, up to a finite critical temperatureTc. Our framework can be applied to a wide class of spin systems and Bell inequalities, to study whether nonlocality occurs naturally in quantum many-body systems close to the ground state. Moreover, we also show that the low-energy spectrum of the Bell operator associated to such systems can be well approximated by the one of a quantum harmonic oscillator, and that spin-squeezed states are optimal in displaying Bell correlations for such Bell inequalities.

Trade-offs in multiparty Bell-inequality violations in qubit networks

Two overlapping bipartite binary input Bell inequalities cannot be simultaneously violated as this would contradict the usual no-signalling principle. This property is known as monogamy of Bell inequality violations and generally Bell monogamy relations refer to trade-offs between simultaneous violations of multiple inequalities. It turns out that multipartite Bell inequalities admit weaker forms of monogamies that allow for violations of a few inequalities at once. Here we systematically study monogamy relations between correlation Bell inequalities both within quantum theory and under the sole assumption of no signalling. We first investigate the trade-offs in Bell violations arising from the uncertainty relation for complementary binary observables, and exhibit several network configurations in which a tight trade-off arises in this fashion. We then derive a tight trade-off relation which cannot be obtained from the uncertainty relation showing that it does not capture monogamy entirely. The results are extended to Bell inequalities involving different number of parties and find applications in device-independent secret sharing and device-independent randomness extraction. Although two multipartite Bell inequalities may be violated simultaneously, we show that genuine multi-party non-locality, as evidenced by a generalised Svetlichny inequality, does exhibit monogamy property. Finally, using the relations derived we reveal the existence of flat regions in the set of quantum correlations.

Closing the detection loophole in multipartite Bell experiments with a limited number of efficient detectors

The problem of closing the detection loophole in Bell tests is investigated in the presence of a limited number of efficient detectors using emblematic multipartite quantum states. To this end, a family of multipartite Bell inequalities is introduced basing on local projective measurements conducted by $N-k$ parties and applying a $k$-party Bell inequality on the remaining parties. Surprisingly, we find that most of the studied pure multipartite states involving e.g. cluster states, the Dicke states, and the Greenberger-Horne-Zeilinger states can violate our inequalities with only the use of two efficient detectors, whereas the remaining detectors may have arbitrary small efficiencies. We believe that our inequalities are useful in Bell experiments and device-independent applications if only a small number of highly efficient detectors are in our disposal.