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Optimal free-will on one side in reproducing the singlet correlation

Published Aug 15, 2016• 0 citations
OpenAlex

Abstract

Bell’s theorem teaches us that there are quantum correlations that can not be simulated by just shared randomness (Local Hidden variable). There are some recent results which simulate singlet correlation by using either 1 cbit or a binary (no-signaling) correlation which violate Bell’s inequality maximally. But there is one more possible way to simulate quantum correlation by relaxing the condition of independency of measurement on shared randomness. Recently, MJW Hall showed that the statistics of singlet state can be generated by sacrificing measurement independence where underlying distribution of hidden variables depend on measurement direction of both parties [Phys. Rev. Lett.105 250404 (2010)]. He also proved that for any model of singlet correlation, 86 % measurement independence is optimal. In this paper, we show that 59 % measurement independence is optimal for simulating singlet correlation when the underlying distribution of hidden variables depend only on measurements of one party. We also show that a distribution corresponding to this optimal lack of free will, already exists in literature which first appeared in the context of detection efficiency loophole. I.

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Bell’s theorem teaches us that there are quantum correlations that can not be simulated by just shared randomness (Local Hidden variable). There are some recent results which simulate singlet correlation by using either 1 cbit or a binary (no-signaling) correlation which violate Bell’s inequality maximally. But there is one more possible way to simulate quantum correlation by relaxing the condition of independency of measurement on shared randomness. Recently, MJW Hall showed that the statistics of singlet state can be generated by sacrificing measurement independence where underlying distribution of hidden variables depend on measurement direction of both parties [Phys. Rev. Lett.105 250404 (2010)]. He also proved that for any model of singlet correlation, 86 % measurement independence is optimal. In this paper, we show that 59 % measurement independence is optimal for simulating singlet correlation when the underlying distribution of hidden variables depend only on measurements of one party. We also show that a distribution corresponding to this optimal lack of free will, already exists in literature which first appeared in the context of detection efficiency loophole. I.
Bell's theorem teaches us that there are quantum correlations that cannot be simulated by just shared randomness (local hidden variable). There are some recent results which simulate the singlet correlation by using either 1 bit or a binary (no-signaling) correlation which violates Bell's inequality maximally. But there is one more possible way to simulate quantum correlation by relaxing the condition of independency of measurement on shared randomness. Recently, Hall showed that the statistics of a singlet state can be generated by sacrificing measurement independence where underlying distribution of hidden variables depends on measurement directions of both parties (Hall 2010 Phys. Rev. Lett. 105 250404). He also proved that for any model of singlet correlation, 86% measurement independence is optimal. In this paper, we show that 59% measurement independence is optimal for simulating the singlet correlation when the underlying distribution of hidden variables depends only on the measurements of one party. We also show that a distribution corresponding to this optimal lack of free will already exists in the literature which first appeared in the context of detection efficiency loophole (Gisin and Gisin 1999 Phys. Lett. A 323–7).
Bell inequalities or Bell-like experiments are supposed to test hidden variable theories based on three intuitive assumptions: determinism, locality and measurement independence. If one of the assumptions of Bell inequality is properly relaxed, the probability distribution of the singlet state, for example, can be reproduced by a hidden variable model. Models that deal with the relaxation of some condition above, with more than one hidden variable, have been studied in the literature nowadays. In this work the relation between the number of hidden variables and the degree of relaxation necessary to reproduce the singlet correlations is investigated. For the examples studied, it is shown that the increase of the number of hidden variables does not allow for more efficiency in the reproduction of quantum correlations.
Bell's theorem rests on three assumptions: locality, determinism, and measurement independence. Experiments violate Bell inequalities, so at least one assumption must be abandoned. The first two exits are well explored; the third — allowing lawful, partial correlation between measurement settings and hidden variables — is usually dismissed as "superdeterminism" and assumed to require total conspiracy. This paper computes what that third exit actually costs. Hall (2010) priced the bipartite case exactly: 13.81% of measurement independence must be surrendered to reproduce singlet-state correlations. This work extends that program to multipartite GHZ/Mermin scenarios and computes, via exact linear programming, the minimum measurement dependence required to reproduce the full faithful statistics — every correlator and every marginal — for 3 through 13 parties. The floors climb a lawful staircase: 1/3, 2/5, 4/9, 8/17, 16/33, 32/65, all obeying the closed form F = R/(2(R+1)) with R the Mermin violation ratio. A proven universal ceiling shows the required dependence never exceeds 1/2 at any experiment size: quantum statistics never force total superdeterminism. The optimal hidden-variable distributions take a closed physical form — uniform measures on the ground states of the prepared state's frustrated stabilizer Hamiltonian — confirmed out-of-sample on cluster states. The result answers a question Hall posed and left open in 2011, and supplies an answer key against which every published lawful-measurement-dependence program (invariant-set theory, cellular-automaton interpretations, future-input dynamics, all-at-once retrocausality, and the displacement-condensate framework of the present authors) is graded in its own formalism. Operationally, the floors are counterfeiting thresholds for multipartite device-independent certificates. Every exact value is machine-verified on independent hardware and reproducible in under one minute from the linked repository: github.com/RandomInternetPreson/moire-phase-space-sampler (DST_Bell_MI directory). Also available at ai.viXra.org/abs/2607.0049.Version 2 (Oct 2026): closed form proved for all n; cluster-core census with exact certificates; scripts and run logs added.
The essence of Bell’s theorem is that, in general, quantum statistics cannot be reproduced by a local hidden variable (LHV) model. This impossibility is strongly manifested when statistics collected by measuring certain local observables on a singlet state, violates the Bell inequality. In this work, we search for local POVMs with binary outcomes for which an LHV model can be constructed for a singlet state. We provide various subsets of observables for which an LHV model can be provided for singlet statistics.
The derivation of Bell inequalities requires an assumption of measurement independence, related to the amount of free will experimenters have in choosing measurement settings. Violation of these inequalities by singlet state correlations brings this assumption into question. A simple measure of the degree of measurement independence is defined for correlation models, and it is shown that all spin correlations of a singlet state can be modeled via giving up just 14% of measurement independence. The underlying model is deterministic and no signaling. It may thus be favorably compared with other underlying models of the singlet state, which require maximum indeterminism or maximum signaling. A local deterministic model is also given that achieves the maximum possible violation of the well-known Bell-Clauser-Horne-Shimony-Holt inequality, at a cost of only 1/3 of measurement independence.
Abstract We propose a novel class of local deterministic models to explain the correlations observed in entangled singlet quantum states. In these models, hidden variables influence both the entangled pair preparation and the measurement settings, thereby relaxing the measurement independence assumption used in Bell’s theorem. Our model accurately reproduces the expected quantum results and demonstrates that freely chosen measurement settings do not necessarily preclude the possibility of hidden variables inducing correlations between the settings and outcomes. This approach reveals an intricate relationship between local determinism and quantum mechanics, showing that local hidden variable theories can be consistent with the predictions of quantum theory. This result calls into question the conventional interpretation, based on Bell’s inequalities, that local theories are fundamentally incompatible with quantum mechanics.