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Simplest proof of Bell’s inequality - Lorenzo Maccone

Simplest proof of bell s inequality - LorenzoMacconeBell s theorem is a fundamental result in quantum mechanics: it discrim-inates between quantum mechanics and all theories where probabilities inmeasurement results arise from the ignorance of pre-existing local proper-ties. We give an extremely simple proof of bell s inequality : a single figuresuffices. This simplicity may be useful in the unending debate of what ex-actly the bell inequality means, since the hypothesis at the basis of the proofbecome extremely transparent. It is also a useful didactic tool, as the Bellinequality can be explained in a single intuitive IntroductionEinstein had a dream. He believed quantum mechanics was an incomplete de-scription of reality [1] and that its completion might explain the troublesomefundamental probabilities of quantum mechanics as emerging from some hid-den degrees of freedom: probabilities would arise because of our ignoranceof these hidden variables.

Simplest proof of Bell’s inequality - Lorenzo Maccone Bell’s theorem is a fundamental result in quantum mechanics: it discrim-inates between quantum mechanics and all theories where probabilities in

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Transcription of Simplest proof of Bell’s inequality - Lorenzo Maccone

1 Simplest proof of bell s inequality - LorenzoMacconeBell s theorem is a fundamental result in quantum mechanics: it discrim-inates between quantum mechanics and all theories where probabilities inmeasurement results arise from the ignorance of pre-existing local proper-ties. We give an extremely simple proof of bell s inequality : a single figuresuffices. This simplicity may be useful in the unending debate of what ex-actly the bell inequality means, since the hypothesis at the basis of the proofbecome extremely transparent. It is also a useful didactic tool, as the Bellinequality can be explained in a single intuitive IntroductionEinstein had a dream. He believed quantum mechanics was an incomplete de-scription of reality [1] and that its completion might explain the troublesomefundamental probabilities of quantum mechanics as emerging from some hid-den degrees of freedom: probabilities would arise because of our ignoranceof these hidden variables.

2 His dream was that probabilities in quantummechanics might turn out to have the same meaning as probabilities in clas-sical thermodynamics, where they refer to our ignorance of the microscopicdegrees of freedom ( the position and velocity of each gas molecule):he wrote, the statistical quantum theory would, within the framework offuture physics, take an approximately analogous position to the statisticalmechanics within the framework of classical mechanics [2].A decade after Einstein s death, John bell [3, 4] shattered this dream inthe worst possible way from Einstein s point of view [4]: any completion ofquantum mechanics with hidden variables would be incompatible with rela-tivistic causality! The essence of bell s theorem is that quantum mechanicalprobabilities cannot arise from the ignorance oflocalpre-existing other words, if we want to assign pre-existing (but hidden) properties toexplain probabilities in quantum measurements, these properties must benon-local.

3 This non-locality is of the worst possible kind: an agent with ac-cess to the non-local variables would be able to transmit information instantlyto a distant location, thus violating relativistic causality and awakening thenastiest temporal paradoxes [5].1 Modern formulations of quantum mechanics must incorporate bell s resultat their core: either they refuse the idea that measurements uncover pre-existing properties, or they must make use of non-local properties. In thelatter case, they must also introduce some censorship mechanism to pre-vent the use of hidden variables to transmit information. An example of thefirst formulation is the conventional Copenhagen interpretation of quan-tum mechanics, which states that the properties arise from the interactionbetween the quantum system and the measurement apparatus, they are notpre-existing: unperformed experiments have no results [6].

4 An example ofthe second formulation is the de Broglie-Bohm interpretation of quantummechanics that assumes that particle trajectories are hidden variables (they exist independently of position measurements). bell s result is at the core of modern quantum mechanics, as it elucidates thetheory s precarious equilibrium with relativistic causality. It has spawned animpressive amount of research. However, it is often ignored in basic quan-tum mechanics courses since traditional proofs of bell s theorem are rathercumbersome and often overburdened by philosophical considerations. Herewe give an extremely simple graphical proof of Mermin s version [7, 8] ofBell s theorem. The simplicity of the proof is key to clarifying all the the-orem s assumptions, the identification of which generated a large debate inthe literature ( see [9]).

5 2 bell s theoremLet us define a local theory as a one where the outcomes of an experimenton a system are independent of the actions performed on a different systemwhich has no causal connection with the first. For example, the temperatureof this room is independent on whether I choose to wear purple socks s relativity provides a stringent condition for causal connections: iftwo events are outside their respective light cones, there cannot be any causalconnection among us define a counterfactual theory [10, 11] as one whose experimentsuncover properties that are pre-existing. In other words, in a counterfactualtheory it is meaningful to assign a property to a system ( the positionof an electron) independently of whether the measurement of such property2is carried out.

6 [Sometime this counterfactual definiteness property is alsocalled realism , but it is best to avoid such philosophically laden term toavoid misconceptions]. bell s theorem can be phrased as quantum mechanics cannot be both localand counterfactual . A logically equivalent way of stating it is quantummechanics is either non-local or non-counterfactual .To prove this theorem, bell provided an inequality (referring to correlationsof measurement results) that is satisfied by all localandcounterfactual the-ories. He then showed that quantum mechanics violates this inequality , andhence cannot be local and experiments [12] performed to date have shown that bell inequalities areviolated, suggesting that our world cannot be both local and , it should be noted that no experiment up to now has been able totest bell inequalities rigorously, because additional assumptions are requiredto take care of experimental imperfections.

7 These assumptions are all quitereasonable, so that only conspiratorial alternatives to quantum mechanics(where experimental imperfections are fine-tuned to the properties of the ob-jects [13]) have yet to be ruled out. In the next couple of years the definitiveBell inequality experiment will be performed: many research groups world-wide are actively pursuing we want to be extremely pedantic in enumerating the hypothesis at thebasis of bell s theorem, we must also request1. that our choice of which experiment to perform is independent ofthe properties of the object to be measured (technically, freedom ofchoice or no super-determinism [4]): , if we decided to measurethe color of red objects only, we would falsely conclude that all objectsare red;2.

8 That future outcomes of the experiment do not influence which appara-tus settings were previously chosen [14] (whereas clearly the apparatussettings will influence the outcomes): a trivial causality requirement(technically, measurement independence ).These two hypothesis are usually left implicit because science would be im-possible without proof of bell s theoremWe use the bell inequality proposed by Preskill [8], following Mermin s sug-gestion [7]. Suppose we have two identical objects, namely they have the sameproperties. Suppose also that these properties are predetermined (counter-factual definiteness) and not generated by their measurement, and that thedetermination of the properties of one object will not influence any propertyof the other object (locality).

9 We will only need three propertiesA,B, andCthat can each take two values: 0 and 1 . For example, if the objects are coins, thenA=0 might meanthat the coin is gold andA=1 that the coin is copper (propertyA,material),B=0means the coin is shiny andB=1 it is dull (propertyB, texture), andC=0 means the coin is large andC=1 it is small (propertyC, size).Suppose I do not know the properties because the two coins are a gift intwo wrapped boxes: I only know the gift is two identical coins, but I do notknow whether they are two gold, shiny, small coins(A=0,B=0,C=1)ortwo copper, shiny, large coins(1,0,0)or two gold, dull, large coins(1,1,0),etc. I do know that the properties exist (namely, they are counterfactualand predetermined even if I cannot see them directly) and they are local(namely, acting on one box will not change any property of the coin in theother box: the properties refer separately to each coin).

10 These are quitereasonable assumptions for two coins! My ignorance of the properties is ex-pressed through probabilities that represent either my expectation of findinga property (Bayesian view), or the result of performing many repeated ex-periments with boxes and coins and averaging over some possibly hiddenvariable, typically indicated with the letter [4], that determines the prop-erty (frequentist view) [6]. For example, I might say the gift bearer will giveme two gold coins with a 20% probability (he is stingy, but not always). bell s inequality refers to the correlation among measurement outcomes ofthe properties: callPsame(A,B)the probability that the propertiesAofthe first object andBof the second are the same:AandBare both 0(the first coin is gold and the second is shiny) or they are both 1 (thefirst is copper and the second is dull).


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