We determine the least Greenberger-Horne-Zeilinger (GHZ) extractability compatible with a Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell score for every number of parties greater than two. Extractability is the largest squared overlap with a GHZ state obtainable by separate local quantum channels. Between the biseparable bound and the quantum maximum, the exact minimum is the affine interpolation from one half to one. The bound holds for normal states on tensor products of arbitrary local dimension and arbitrary binary measurements. We settle the previously unresolved range of six or more parties with an analytic proof that is uniform from four parties onward. We also determine the exact minimum when every local system is a qubit, and show that it lies strictly above the unrestricted bound at every interior score. One qutrit and qubits at all remaining parties attain every point of the bound with measurements fixed as the score varies. For at least four parties and strict interior scores, we classify all states attaining the bound with the minimum product of local support dimensions. For at least four parties on one qutrit and qubits elsewhere, we also prove that, when the extractability excess above the affine minimum is small, the trace-norm distance from the attaining mixture, up to local unitaries, is bounded by a constant times the square root of that excess. The exponent $1/2$ is optimal, and the constants are independent of the number of parties on every fixed interior interval of normalized scores.
We determine the least Greenberger-Horne-Zeilinger (GHZ) extractability compatible with a Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell score for every number of parties greater than two. Extractability is the largest squared overlap with a GHZ state obtainable by separate local quantum channels. Between the biseparable bound and the quantum maximum, the exact minimum is the affine interpolation from one half to one. The bound holds for normal states on tensor products of arbitrary local dimension and arbitrary binary measurements. We settle the previously unresolved range of six or more parties with an analytic proof that is uniform from four parties onward. We also determine the exact minimum when every local system is a qubit, and show that it lies strictly above the unrestricted bound at every interior score. One qutrit and qubits at all remaining parties attain every point of the bound with measurements fixed as the score varies. For at least four parties and strict interior scores, we classify all states attaining the bound with the minimum product of local support dimensions. For at least four parties on one qutrit and qubits elsewhere, we also prove that, when the extractability excess above the affine minimum is small, the trace-norm distance from the attaining mixture, up to local unitaries, is bounded by a constant times the square root of that excess. The exponent 1/2 is optimal, and the constants are independent of the number of parties on every fixed interior interval of normalized scores.
We analytically characterize the maximal quantum violation of the extended Mermin-Ardehali-Belinskii-Klyshko (EMABK) family of inequalities by n-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states. We develop a correlation-tensor approach in which the Bell value is expressed as the Frobenius inner product of the quantum correlation tensor and an effective coefficient tensor. A rank constraint on the latter, together with the von Neumann trace inequality, yields an upper bound governed by the two largest singular values of the reshaped correlation-tensor. For generalized GHZ states, we determine the complete singular spectrum and obtain a piecewise analytic upper bound. We then construct two complementary measurement strategies that saturate the bound for EMABK throughout the entire parameter range. The first is a purely MABK strategy in which all measurement directions lie in the equatorial plane of the Bloch sphere. The second is a hybrid strategy that recursively combines lower-order MABK anti-diagonal operators with the fully-σz tensor product. The resulting piecewise analytic expression recovers the standard MABK Tsirelson bound 2(n−1)/2 in the maximally entangled limit and approaches the classical local-hidden-variable bound in the product-state limits. It further shows that every entangled generalized GHZ state exhibits a strict quantum-classical separation under the EMABK inequality, thereby eliminating the nonviolation region of the standard MABK inequality in the partially entangled regime.
Bell nonlocality provides a device-independent (DI) way to certify quantum randomness, based on which true random numbers can be extracted from the observed correlations without detail characterizations on devices for quantum state preparation and measurement. However, the efficiency of current strategies for DI randomness certification is still heavily constrained when it comes to non-maximal Bell values, especially for multiple parties. Here, we present a family of multipartite Bell inequalities that allows to certify optimal quantum randomness and self-test GHZ (Greenberger-Horne-Zeilinger) states, which are inspired from the stabilizer group of the GHZ state. Due to the simple representation of stabilizer group for GHZ states, this family of Bell inequalities is of simple structure and can be easily expanded to more parties. Compared with the Mermin-type inequalities, this family of Bell inequality is more efficient in certifying quantum randomness when non-maximal Bell values achieved. Meanwhile, the general analytical upper bound for the Holevo quantity is presented, and achieves better performance compared with the MABK (Mermin-Ardehali-Belinskii-Klyshko) inequality, Parity-CHSH (Clauser-Horne-Shimony-Holt) inequality and Holz inequality at N=3, which is of particular interests for experimental researches on DI quantum cryptography in quantum networks.
A local deterministic model can reproduce Bell-inequality violations only if the hidden state depends on the measurement settings. Hall found the minimum such dependence for the singlet state [Phys. Rev. Lett. 105, 250404 (2010)] and posed the corresponding problem for the Mermin state [Phys. Rev. A 84, 022102 (2011)]. This paper solves it, for the n-party GHZ-Mermin scenario at every n≥3, with the marginals reproduced as well as the correlators. The minimum measurement-dependence fraction is Fmin(n)=R/[2(R+1)], with R=2⌊(n−1)/2⌋ the Mermin violation ratio: 1/3,2/5,4/9,8/17,…, always below 1/2. The lower bound follows from Mermin's inequality by a general overlap bound; the upper bound is an explicit model, uniform on the hidden-variable classes that violate the fewest constraints, with closed-form counts; a reduction theorem shows that for Mermin scenarios the marginals impose no additional dependence. The Mermin values are also certified by integer-arithmetic enumeration through n=22. The formula was stated as a conjecture in the first version of this paper and was first proved, by a different route, by Wang, Yao and Chen; the proof here was obtained independently and the two agree. Every minimal contradiction of size k among stabilizer constraints has correlator-only minimum 1/(k−1). With the marginals included the minimum is the same on every core of the C5 ring cluster, on the size-4 and size-5 cores of C4 and on every four-core of C6, but larger on the size-6 and size-7 cores of C4 and on 112 of the 1,920 five-cores of C6, with exact certificates on both sides in every case.
Nonlocal tests on multi-partite quantum correlations form the basis of protocols that certify randomness in a device-independent (DI) way. Such correlations admit a rich structure, making the task of choosing an appropriate test difficult. For example, extremal Bell inequalities are tight witnesses of nonlocality, but achieving their maximum violation places constraints on the underlying quantum system, which can reduce the rate of randomness generation. As a result there is often a trade-off between maximum randomness and the amount of violation of a given Bell inequality. Here, we explore this trade-off for more than two parties. More precisely, we study the maximum amount of randomness that can be certified by correlations with a particular violation of the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality. For any even number of parties, we find that maximum randomness cannot occur beyond a threshold quantum violation, which increases with the number of parties, and we give a conjectured form of the maximum randomness in terms of the MABK value. We also show that maximum randomness can be obtained for any MABK violation for odd numbers of parties. To obtain our results, we derive new families of Bell inequalities certifying maximum randomness from a technique for randomness certification, which we call "expanding Bell inequalities''. Our technique allows a bipartite Bell expression to be used as a seed, and transformed into a multi-partite Bell inequality tailored for randomness certification, showing how intuition learned in the bipartite case can find use in more complex scenarios.
Self-testing is a device-independent technique that allows for verification of unknown quantum systems only when observed correlations achieve some quantum supremum. In this paper, we present self-testing schemes for a class of multipartite Bell inequalities with a specific structure, which we refer to as the unit. Our research is based on the graph-theoretic approach, where correlations are characterized via Lov\'asz theta body in the framework of exclusivity graphs. The derived statements are subsequently applied to prove the self-testability of N-partite Mermin-Ardehali-Belinski\ifmmode \check{i}\else \v{i}\fi{}-Klyshko (MABK) family of Bell inequalities. In particular, we take special cases to demonstrate that the maximal violation of the multipartite MABK inequality is a self-test for the corresponding multipartite Greenberger-Horne-Zeilinger (GHZ) state. Moreover, we propose a device-independent verification algorithm for distributed blind quantum computation in the noisy intermediate-scale quantum era combined with multipartite GHZ state self-testing, where multiple quantum servers are involved and work in collaboration to reduce the burden on a single server.
We introduce a real-valued functional I(n1,n2) of four three-qubit correlations that turns the Greenberger-Horne-Zeilinger (GHZ) algebraic paradox into a quantitative witness of genuine tripartite entanglement. For every three-qubit state ρ and every pair of directions, ∣I(n1,n2;ρ)∣≤2, with equality if and only if n1⊥n2 and ρ is locally unitarily equivalent to the GHZ state. On the five-parameter Acín family, I(x^,y^) depends only on λ0λ4 and is maximised by the GHZ state; for the W state I(x^,y^)=0 and max∣IW∣=35/27. Maximising over independent local orthonormal frames yields a local-unitary invariant measure EGHZ∈[0,1], computable from four three-body Pauli correlators, that equals one exactly on the GHZ orbit and takes the values 35/54 on the W state and 1/2 on product states. We show that EGHZ is not convex: an equal mixture of three biseparable states has EGHZ=14/27, larger than the average 1/2 of its components. We also relate EGHZ to GHZ fidelity witnesses, discuss its behaviour under local operations and classical communication, and outline a generalisation to three qudits via Heisenberg-Weyl operators.