A robust mathematical model for a loophole-free Clauser–Horne experiment

Bierhorst, Peter

doi:10.1088/1751-8113/48/19/195302

Cited by 34 publications

(39 citation statements)

References 30 publications

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“…33 In some cases it is possible to transform a general game into a win/lose game by postselecting the trials that take the maximum and minimum value. 2,16 In that situation, it would be possible to apply the tight bounds for win/lose games. Techniques sometimes referred to as 'speeding up time' 2,28 can analogously be used in conjunction with this refined bound.…”

Section: General Gamesmentioning

confidence: 99%

“…Hence, in the case of a Bell experiment, a small P value can be regarded as strong evidence against the hypothesis that the experiment was governed by an arbitrary LHVM. There is extensive literature regarding the methods for evaluating the P value in Bell experiments [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and discussions regarding the analysis of concrete experiments and loopholes. [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] Previous approaches to obtain such P values known from the literature can be roughly divided into two categories.…”

Section: Introductionmentioning

confidence: 99%

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(Nearly) optimal P values for all Bell inequalities

Elkouss

Wehner

2016

npj Quantum Inf

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A key objective in conducting a Bell test is to quantify the statistical evidence against a local-hidden variable model (LHVM) given that we can collect only a finite number of trials in any experiment. The notion of statistical evidence is thereby formulated in the framework of hypothesis testing, where the null hypothesis is that the experiment can be described by an LHVM. The statistical confidence with which the null hypothesis of an LHVM is rejected is quantified by the so-called P value, where a smaller P value implies higher confidence. Establishing good statistical evidence is especially challenging if the number of trials is small, or the Bell violation very low. Here, we derive the optimal P value for a large class of Bell inequalities. What is more, we obtain very sharp upper bounds on the P value for all Bell inequalities. These values are easily computed from the experimental data, and are valid even if we allow arbitrary memory in the devices. Our analysis is able to deal with imperfect random number generators, and event-ready schemes, even if such a scheme can create different kinds of entangled states. Finally, we review requirements for sound data collection, and a method for combining P values of independent experiments. The methods discussed here are not specific to Bell inequalities. For instance, they can also be applied to the study of certified randomness or to tests of noncontextuality.

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Section: General Gamesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

(Nearly) optimal P values for all Bell inequalities

Elkouss

Wehner

2016

npj Quantum Inf

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“…Now consider the following result, which is proved as the second proposition in Appendix C of [51] Proposition. Let (T i ) n i=1 be a sequence of random variables taking values in the set {0, 1}.…”

Section: Statistical Procedures and Proof Of Claimsmentioning

confidence: 97%

Chained Bell Inequality Experiment with High-Efficiency Measurements

et al. 2017

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We report correlation measurements on two 9 Be + ions that violate a chained Bell inequality obeyed by any local-realistic theory. The correlations can be modeled as derived from a mixture of a local-realistic probabilistic distribution and a distribution that violates the inequality. A statistical framework is formulated to quantify the local-realistic fraction allowable in the observed distribution without the fair-sampling or independent-and-identical-distributions assumptions. We exclude models of our experiment whose local-realistic fraction is above 0.327 at the 95 % confidence level. This bound is significantly lower than 0.586, the minimum fraction derived from a perfect ClauserHorne-Shimony-Holt inequality experiment. Furthermore, our data provides a device-independent certification of the deterministically created Bell states.Recently several groups have reported loophole-free tests of local realism with Bell's theorem [1], rejecting with high confidence theories of local realism [2][3][4]. While these experiments falsify the idea that nature obeys local realism, they are limited in the extent to which their data differs from local realism. Chained Bell inequality (CBI) [5] experiments can show greater departures from local realism in the following sense: Elitzur, Popescu, and Rohrlich [6] described a model of the distribution of outcomes measured from a quantum state as a mixture of a local-realistic distribution, which obeys Bell's inequalities, and another distribution that does not. Following their convention, we call these distributions "local" and "non-local." According to Ref. [6], a probability distribution P for the outcomes of an experiment can be written aswhere P L represents a local joint probability distribution (a "local part") and P N L represents a non-local distribution, with p local as the weight of the local component bound by 0 ≤ p local ≤ 1. For an ideal Clauser-HorneShimony-Holt (CHSH) Bell inequality experiment where two physical systems (usually particles) are jointly measured with four different measurement settings [7], the lowest attainable upper bound on the local content p local in any quantum distribution is ∼ 0.586 [8,9]. In principle, this bound can be lowered to zero by using a chained Bell inequality experiment.As indicated in Fig. 1.(a), CBI experiments are a generalization of a CHSH-type experiment. During each trial, a source that may be treated as a "black box" emits two systems labeled a and b, respectively. The experimentalist records the measurement outcomes after choosing a pair of measurements to perform separately on a and b. We use the symbols a k , b l to denote the respective measurement settings and a k b l for the pair. The latter is usually simply referred to as "the settings" or "the setting pairs". There is a hierarchy in which the Bell test SourceMeasurement:Illustration of a Bell inequality experiment. A source emits two systems a and b, here two 9 Be + ions. After choosing measurement settings a k and b l , the experiment implements Hilbert-space ...

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“…Based on the works [40,57], we present the first statistical analysis with the following three key features, all of which are essential for a photonic Bell state with current technology:…”

Section: Statistical Significance and Run Timementioning

confidence: 99%

Requirements for a loophole-free photonic Bell test using imperfect setting generators

et al. 2016

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Experimental violations of Bell inequalities are in general vulnerable to so-called "loopholes." In this work, we analyse the characteristics of a loophole-free Bell test with photons, closing simultaneously the locality, freedom-of-choice, fair-sampling (i.e. detection), coincidence-time, and memory loopholes. We pay special attention to the effect of excess predictability in the setting choices due to non-ideal random number generators. We discuss necessary adaptations of the CH/Eberhard inequality when using such imperfect devices and -using Hoeffding's inequality and Doob's optional stopping theorem -the statistical analysis in such Bell tests.

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A robust mathematical model for a loophole-free Clauser–Horne experiment

Cited by 34 publications

References 30 publications

(Nearly) optimal P values for all Bell inequalities

(Nearly) optimal P values for all Bell inequalities

Chained Bell Inequality Experiment with High-Efficiency Measurements

Requirements for a loophole-free photonic Bell test using imperfect setting generators

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