A local hidden-variable model is constructed for the postselected correlations of nonmaximally entangled photon pairs in the states |Phi_r> and |Psi_r>, continuous in r over [-1,1] from the product state to the maximally entangled state. Its root-mean-square deviations from the quantum correlations and from the coincidence probabilities, averaged over a uniform grid of polarizer angles, are 0.0322 and 0.0233, smaller than those of the sigmoid-threshold model of Ref. [14] by factors of 2.7 to 3.5. The source emits a photon pair with a probability that depends on the hidden variable and on r; each station detects with a probability that depends on the local hidden variable and on the local polarizer angle. This dependence violates the fair-sampling assumption and exploits the detection loophole. On the postselected sample the model violates the CHSH inequality for |Phi_r>, reaching 2*sqrt(2) at |r|=1, with single-detector efficiency ≈ 85%, outside the assumptions of both the CHSH inequality and Mermin's 82.8% efficiency bound. The CH--Eberhard parameter, which does not assume fair sampling, satisfies J<0 over the full entanglement range.
The Greenberger–Horne–Zeilinger (GHZ) puzzle has been used to study quantum nonlocality and provide an all-or-nothing, no-go theorem for local hidden-variable models. Recent experiments using coincident-detected entangled photons prepared in a three-particle GHZ state have been used to test quantum nonlocality, but fail to rule out local realism due to a reliance on the fair-sampling hypothesis and insufficient detection efficiency. This paper describes a physically motivated local hidden-variable model based on amplitude-threshold detection that is capable of producing similar results. Detection efficiencies for the model are within the bounds permitted for local realism and, interestingly, exhibit statistical correlations between detectors, even when the detection events are spacelike separated. Increasing the detection threshold improves agreement with the ideal quantum predictions at the cost of decreased detection efficiency. A curious emergent feature of the model is that detection efficiencies may depend upon which observables are chosen for measurement.
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.
A theoretical and numerical analysis of a Bell-type experiment is presented, where linear polarization measurements on nonmaximally entangled two-qubit states are modeled through a sigmoid-based threshold mechanism that violates the fair-sampling assumption required by the CHSH inequality. Numerical simulations characterize how closely this local construction can reproduce quantum-like correlations in the postselected detected sample, for both the | Φ r 〉 and | Ψ r 〉 states over a range of entanglement parameters. For the tested configurations, the model produces CHSH-type violations only in the postselected detected sample through a setting-dependent detection mechanism, operating outside the assumptions of both the CHSH inequality and Mermin’s 82.8% efficiency bound. The model is not in conflict with loophole-free Bell experiments, which specifically close the detection loophole exploited here. The CH–Eberhard analysis in the one-detector-per-side configuration yields J < 0 throughout the full range of entanglement parameters, as required for a strictly local model. The model is further extended with a local adaptive-threshold mechanism implementing one-sided memory. A paired comparison shows that this specific memory rule produces a systematic and reproducible shift in the CH–Eberhard parameter J relative to the fixed-threshold model, while leaving the CH–Eberhard analysis qualitatively unchanged under the tested conditions. The results characterize how a local, setting-dependent postselection mechanism shapes Bell-test statistics across a range of entanglement parameters, within the specific configurations and threshold rules tested.
We introduce for a general correlation scenario a new simulation model, a local quasi hidden variable (LqHV) model, where locality and the measure-theoretic construction inherent to a local hidden variable (LHV) model are preserved but positivity of a simulation measure is dropped. We specify a necessary and sufficient condition for LqHV modelling and, based on this, prove that every quantum correlation scenario admits a LqHV simulation. Via the LqHV approach, we construct analogs of Bell-type inequalities for an N-partite quantum state and find a new analytical upper bound on the maximal violation by an N-partite quantum state of S1 × ⋯ × SN-setting Bell-type inequalities – either on correlation functions or on joint probabilities and for outcomes of an arbitrary spectral type, discrete or continuous. This general analytical upper bound is expressed in terms of the new state dilation characteristics introduced in the present paper and not only traces quantum states admitting an S1 × ⋯ × SN-setting LHV description but also leads to the new exact numerical upper estimates on the maximal Bell violations for concrete N-partite quantum states used in quantum information processing and for an arbitrary N-partite quantum state. We, in particular, prove that violation by an N-partite quantum state of an arbitrary Bell-type inequality (either on correlation functions or on joint probabilities) for S settings per site cannot exceed (2S − 1)N − 1 even in case of an infinite dimensional quantum state and infinitely many outcomes.
Local hidden-variable models can reproduce Bell violations if settings depend on hidden variables. Using total-variation distance, we determine the minimum dependence needed to reproduce all outcome probabilities for Pauli X and Y measurements on n-qubit Greenberger-Horne-Zeilinger states, for every n ≥ 3, visibility and coherence phase. For odd n, the larger magnitude of the real and imaginary coherence components sets the cost; for even n, the smaller can raise it. States with identical, Bell-violating Mermin data can thus have different costs.