Bell's Theorem From a Weak Assumption on Correlations - Alejandro Cabrera, Edson de Faria, Enrique...

7/27/2019 Bell's Theorem From a Weak Assumption on Correlations - Alejandro Cabrera, Edson de Faria, Enrique Pujals, Char

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arXiv:1311.6112v1

[quant-ph]24Nov2013

Bells Theorem from a weak assumption on correlations

Alejandro CabreraInstituto de Matematica, UFRJ, CEP 21941-909, Rio de Janeiro - Rio de Janeiro, Brazil

Edson de FariaInstituto de Matematica e Estatstica, USP, Rua do Matao 1010, SP 05508-090 , Sao Paulo, Sao Paulo, Brazil

Enrique PujalsIMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Rio de Janeiro, Brazil

Charles TresserIBM, P.O. Box 218, Yorktown Heights, NY 10598, U.S.A.

(Dated: November 26, 2013)

We prove a version of Bells Theorem that does not assume Locality. The hypotheses that we useinstead are proven to be weaker than Locality. One of them follows from the impossibility of super-luminal signaling. We prove that a certain correlation (in an EPR situation) is one time differentiableand that very mild further differentiability assumption implies a Bell type contradiction.

We consider sequences of EPRB pairs, i.e., pairs of

spin-

1

2 particles (p, p) whose wave functions spin part isthesinglet state:

= 1

2(|+p |p |p |+p) .

The EPRB name evokes a reformulation by Bohm [1] ofthe seminalEPR paper [2] using spin- 1

2 particles pairs.

Consider a sequence of EPRB pairs (pi, pi). In whatfollows,corresponding measurementsmade by Alice andBob are measurements respectively made by these twoagents on the members pi and pi of any given singletstate pair that they get. By definitionpi is the memberof the ith pair that flies to Alice while pi is the member

of that pair that flies to Bob. Thus Alice measures thenormalized projectionsa = {ai} of the spins of the (pi)salong the vector va at anglea while Bob measures thenormalized projectionsb = {bi}of the spins of the (pi)salong vb . All our vectors belong to some fixed plane, andthe anglecis measured relative to a reference vector v0at angle 0 chosen once and for all in that plane. We shallsetcd c d. Quantum mechanics (QM) teaches usthat the correlationa, b is given by the twisted Maluslaw (TML):

a, b limn

1

n

n

i=1

aibi= cos(ab)

ABell type experiment (i.e, an experiment of the typeconsidered in Bells Theory: see [3] and subsequent pa-pers on the same theme) only provides two sequencesof observed normalized spin projections: a ={ai} andb ={bi}. Predictive Hidden Variables (PHVs) theorieshave often been proposed as a mean to obtain a determin-istic theory for micro-physics. The characteristic prop-erty of PHVs of relevance for us here is the fact that theyare realistic so that observable values have a meaning

before (and in fact independently of) being measured.

Except perhaps for proponents and adopters of PHVstheories, instead of using PHVs as Bell in [3] one tendsnow following Stapp [4] (see also [5] and [6]) to ratheruse one of the weaker realism assumptions that go collec-tively under the name of Classical Microscopic Realism(CMR). CMR tells us that all the observables that couldbe measured have well defined values. The weakest formof CMR needed to discuss Bells Theory is used in[7] and[8]. We will stick to CMR to which we have adapted thefollowing two conventions that go back at least to [3].

Convention 1. We assume that any n quantities thatare not measured, but that exist according to CMR, havethe values that would have been obtained if they had been

jointly measured quantities. We assume here that saidsets ofn quantities can indeed be co-measured.

In relation with Convention 1, we notice that we willonly be concerned in cases when n 2.Convention 2. We assume that the statistical predic-tions (e.g., average values and in particular correlations)are the ones given by QM.

We call the angles used by Alice and Bob for mea-surementsphysical (choices of) angles. To mark the cor-responding previously defined sequences a and b so asto also be prepared to denote other possibilities, we use

(a, b) (a, b) where the left hollow circle (resp. righthollow circle) in the superscript ofc in{a, b} marks thatAlice (resp. Bob) has used a physical angle. We willuse a filled circle instead of a hollow circle in the super-script pair of Alices or Bobs sequence to represent thefact that said sequence is a CRM-dependent sequence ofvalues (inferred to exist from CMR along a non-physicalangle). For instance, besides measurements on both ele-ments of each pair, one may consider the following alter-nate situations:
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AS1a Alice could have chosen another angle of possiblemeasurementa with Bob keeping the angle b, inwhich case they would respectively have obtainedthe sequences a {ai } and b {bi } (ortogether the pair of sequences (a, b)).

AS1b Bob could have chosen another angle of possiblemeasurementb , while Alice kept the angle a, in

which case they would have obtained the pair ofsequences (a, b) .

AS2 Alice and Bob could both have chosen the alter-nate angles a and b , and thus would have ob-tained the pair of sequences of counterfactual val-ues, (a, b) that is used in [9].

The word measured when applied to one sequencedoes not indicate that the corresponding sequence isknown: only effectively the co-measured sequences (a

and b) are are assumed to be known a priori.While the two sequences that make sense together

must carry the same super-scripts, we will also considermixed pairs such as, e.g.,(a, b) (cf. (2), (3) ).

Remark 1. We stress that the sequences which are notmeasured have a physical meaning only when assumingthe CMR hypothesis. We shall continue assuming CMRbut our goal is to prove ab absurdo that CMR makes nophysical sense, at least under some very mild assump-tions.

Recall now that Localitymeans that the outputs (ei-ther measured or inferred to exist by assuming CMR) donot depend on the choice of vector made by the other

observer when the measurements by Alice and Bob arespace-like separated (i.e.,||x|| > c ||t|| for the spaceand time distances where c is the speed of light in vac-uum).

Remark 2. With the set of hypotheses made so far, thereis no reason to expect (as holds true when assuming Lo-cality) that for c {a, b}, c1 1 = c2 2 beyond thecase when1 =2 and1 =2. In particular, eightdifferent sequences could have been generated .

We do not assume Locality here as it is known sinceBells 1964 paper that assuming QM, PHVs, and Localityimplies a contradiction [3] (but CMR instead of PHVs

works as well).

Without further assumption than QM, the TML letsus compute a, b, but when we assume both QM andCMR, more observables values make sense and it followsreadily, using Conventions 1 and 2, that the TML appliesto more pairs so that we have:a, b = cos(ab), a, b = cos(ab),a, b = cos(ab), anda, b = cos(ab)(see also[9]).

Consider now the two inequalities

|x1+x2| + |x3 x4| 2,|x1 x2| + |x3+x4| 2.The set of points in R4 that satisfy both inequalitiesdefines a polytope P I4 R4, where I = [1, +1].Let us recall [10],[11]:Booles Lemma:

| u, x + v, x | + | u, y v, y | 2 (1)where, u, x, v, and y are sequences of numbers in thedoubleton{1, 1} such that the limits defining the 4 cor-relations involved in (1) exist.

Notice that we could apply Booles lemmato all correlations associated to the ntuplesa, b, a, b, a, b, a, b obtained before. Recallhowever that in order to apply that lemma, only fourof these correlations can be used simultaneously. Also,following a tradition that goes back to [12], we onlyconsider correlations of the form

a11 , b22

. Since

all the sequences that we consider in these correlations

have values in{1, +1}, it follows from Booles Lemmaunder standard convergence hypotheses that one has theCHSH inequalities [12]:

| a, b + a, b | + | a, b a, b | 2 (2)| a, b a, b | + | a, b + a, b | 2 (3)

Recall that with the hypotheses that we have made sofar, the correlationsa, b,a, b, anda, bare all unknown to us.

Definition 1. The Restricted effect after cause principle(REACP) states that for any Lorentz observer (LO), andin particular with respect to the time ordering for that

LO, the registered observable value after a physical choiceof angle cannot change because of further choices that aremade after that.

The REACP is formulated and used only for measure-ments (and their registrations) that are performed. Wecame to isolate the REACP as more efficient an hypothe-sis than theeffect after cause principle (EACP) form[7],[8]. In particular we will prove below the REACP to be adirect consequence of Special Relativity (SR), somethingthat we could not prove (so far) for the EACP and thatmakes corrections of[7] and[8] much stronger.

When assuming the REACP we do not ask from CMR-

dependent values the same that is asked from actual val-ues that are generated by measurements because of thefollowing simple observation (in the spirit of the title ofa famous short paper[13]):- What happens may have consequences but what couldhave happened (at the atomic scale at least) has no con-sequence (on the subsequent actual world).- In particular,Only an actual measurement by Alicemay have an effect on the sequence measured by Bob,and vice versa.


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Recall now that in order to use inequalities (2) we needto compute a, b, a, b and a, b, or at leastwe need to know enough about these correlations.

Claim 1. [REACP ] [b =b and a =a].This claim follows from choosing different LOs:

(i) Considering one LO for which the measurements

done by Bob happen before the corresponding onesdone by Alice, the REACP tells us directly that nochange performed by Alice can affect the measure-ment done by Bob which implies thatb =b.

(ii) Considering instead another LO for whom Alicemeasures before Bob on corresponding particles, wegeta =a mutatis mutandis. Q.E.D.

Corollary 1. Assuming QM and CMR it follows fromthe REACP thata, b = a, b = cos(ab) andthata, b = a, b = cos(ab).Remark 3. The REACP is nothing else than the EACPof [7] and[8] (that also yields Corollary 1) restricted toactual measurements. Since the REACP can, purposely,only influence consequences of performed measurements,we observe that it may well be thatb =b ora =a,so we do not know how the correlationa, b behaves(as a function of the angles on which it depends) un-der [QM + CMR + REACP ]. To the contrary, Local-ity implies Corollary 1 and also allows one to computea, b [3]. On the other hand, we prove here thatthe REACP is weaker than Locality(see Theorem2andCorollary2).

Let us denote T [, ]/( ) and I [1, 1].There is a function

C: T4

I4 which represents how the

following correlations depend on choices of quadrupletsof angles = (a, a , b, b) in T4. We have:

C() = (a, b , a, b , a, b , a, b)= ( cos(ab), cos(ab), cos(ab), a, b ()),

and it follows from the CHSH inequalities (2) and (3)that:

T4 , C() P. (4)We are going to prove some weak regularity properties

of the correlationa, b as a function of, and thenalso prove that we get a contradiction if we further makethe mild assumption of a bit more of regularity of thatfunction. In view of the computed and measured correla-tion functions in the context that we are in, assuming onedegree more of differentiability for a correlation that weprove differentiable would appear a priorimuch weakerthan assuming the first degree of differentiability.

Observing that (ab+ab) (ab+ ab) = 0 we canconsider variables1, 2, 3, and 4 such that:

1 = ab, 2 = ab, 3 = ab , 4 = 1 (2+3).

Let us denote by the triplet of angles differences(ab, ab, ab), so that = (1, 2, 3). Also, let:

Q+= {= (1, 2, 3) R3 : 0 i , i= 1, 2, 3} ,

and consider the function F4 : Q+ R defined byF4() = a, b.Claim 2. Suppose [QM+CMR+REACP] and supposealso thatC satisfies (4). Then, F4 is differentiable atO= (0, 0, 0)andDF4(O) = (0, 0, 0).

From (2) and the definition of F4 it follows thatF4(O) =1. Moreover, for close to O, the CHSHinequalities imply:

|F4() + 1| 3 cos 1 cos 2 cos 3.

Dividing by the norm of , we get a quotient that clearlygoes to zero as O. Q.E.D.

We thus have established that F4 is differentiable atO Q+. The typical expectation is that if such a cor-relation function is once differentiable, it is real analytic,but we shall not assume that much on top of the regu-larity that we have just proved.

Theorem 1. Suppose [QM+CMR+REACP], and alsoboth thatC satisfies (4) and thatF4, hencea, b, istwice differentiable at = O as a function restricted toQ+. Then, it would follow that3 = 1.

First, it follows that from general considerations thatunder the present [QM+CMR] assumptions we haveF4(,,) = cos() for any . Now, we illustrate themain point of the proof by assuming that F4 is a degree2 polynomial around = O , whence:

F4 = 1 +

i

cii+

ij

cijij,

(the general case follows readily by using the ele-mentary theory of Taylor polynomials.) We accord-ingly replace the function cos(x) by its order-two ap-proximating polynomial1 + 1

2x2. When assuming

[QM+CMR+REACP], by Claim2all linear coefficientsmust vanish: ci = 0 for i = 1, 2, 3. Next, it is not hardto check that if|z| 1,|w| 1 and|1 z| + |1 w| 2then z = w (cf. [9]). In this way, we thus obtain thatc11 =

1

2 by considering = (x, 0, 0). Analogously, we

obtain c22 = c33 = 12 . Now, taking = (x,x,x) andnoting thatF4 coincides with the cosine on the diagonal,we have:

1

2+

1

2+

1

2+

i


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4

= (0,x,kx) (withx small and kxalso small but fixed);we get:

2 12

x2 +

1

2x2 +c23kx

2

2 +o(x2)

for all values ofk. In particular, canceling 2, dividing byx2 and takingk arbitrarily large, we see that the inequal-

ity holds only whenc23 = 0. The equalitiesc13 = 0 = c23are shown in the same way. Therefore, equation (5) im-plies that 3 = 1. Q.E.D.

Remark 4. Since Theorem 1 only requires functionalregularity when restricted toQ+, it also holds when thecorrelation a, b or its derivatives are allowed to

jump as some of the is (i = 1, 2, 3) changes sign; inparticular, Theorem1can be applied to correlations thatdepends on absolute value. Also observe that in the proofof Theorem1 is only being used that the known three cor-relations are twice differentiable at the origin with firstderivative equal zero.

Theorem 2. The REACP is implied by SR, and morespecifically by the impossibility of sending superluminalmessages.

To prove this statement we will show that the nega-tion of the REACP allows superluminal signaling. Tofix the ideas, we assume that two slots1i and +1i areassociated to each event in a sequence of events labeledby i. The result of an experiment is then marked byputting a slot marker in one of these two slots. Accord-ingly, if the REACP fails, the marker of some slot wouldmove because of later manipulations in the time frame ofsome LO. Even if only some sufficient proportion of slots

would get removed or displaced, trivial error correctingtechniques by repetition of symbols and intertwining ofwords (to better correct bunched errors) would allow tosend YES or NO messages with high probability back-ward in time, which is the basic form of superluminaltransmission. This argument also shows that the nega-tion of the REACP is stronger than the negation of SR,so that the REACP follows from SR. This proves Theo-rem2. Q.E.D.

Corollary 2. The conjunction of the REACP with theregularity hypothesis of Theorem1 is strictly weaker thanLocality when assuming [QM +CRM].

To prove this statement, we first recall that on the onehand it is well known (see, e.g., [14]) that the negationof Locality does not imply the possibility to transmit su-perluminal messages(sequences of symbols carrying rec-ognizable information content). On the other hand, by

Theorem 2 the negation of the REACP allows superlu-minal messages.

It then remains to deal with the regularity part of Corol-lary 2. For that, we notice that Locality permits oneto conclude thata, b = cos(ab) when assuming[QM+CMR] (as explained by Bell in [3]): this formulaprovides much more regularity than needed. Q.E.D.

Conclusion. We have proved here a very strong ver-sion of Bells Theorem assuming SR (that implies theREACP) as well as QM and CMR: the correlation regu-larity near (0, 0, 0) needs to be somewhere between onceand twice differentiable in the positive coordinate sectorto escape a fallacious consequence. Together with [7],[8], and[9] that mentions recent related work by severalother authors, we have now a collection of strong formsof Bells Theory without Locality; altogether there is anincreasing body of work invalidating CMR.

Acknowledgments: This work was made possibleby support from FAPESP through Pro jeto TematicoDinamica em Baixas Dimensoes, Proc. FAPESP2006/03829-2, and also Proc. FAPESP 2012/19995-0.The support of the Palis-Balzans Prize is also gratefullyacknowledged.

[email protected]; [email protected];[email protected];[email protected]

[1] D. Bohm Quantum Theory, (Prentice Hall; New York1951).

[2] A. Einstein, B. Podolsky, and N. Rosen Phys. Rev. 47,777 (1935).[3] J.S. Bell, Physic(Long Island City, NY) 1, 195 (1964).[4] H.P. Stapp, in Symposium on the Foundations of Mod-

ern Physics: 50 years of the Einstein-Podolsky-RosenGedankenexperiment,P. Lahti and P. Mittelstaedt, Eds.(World Scientific, Singapore, 1985), pp. 637-652.

[5] H.P. Stapp,Phys. Rev. 3 D, 1303 (1971).[6] A.J. Leggett, Rep. Prog. Phys. 71, 022001 (2008).[7] C. Tresser, Eur. Phys. J. D 58, 385 (2010).[8] C. Tresser, Eur. Phys J. D 62, 139 (2011).[9] E. de Faria, C. Tresser, Phys. Rev. Lett. 110, 260409

(2013).[10] G. Boole, Phil. Trans. Roy. Soc. (London) 152, 225

(1862).

[11] I. Pitowsky,Phys. Rev. A 64, 4102 (2001).[12] J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt,

Phys. Rev. Lett. 23, 880 (1969).[13] A. Peres, Quantum Theory: Concepts and Methods,

(Kluwer: Dordrecht, 1993).[14] P. Eberhard, Nuovo Cimento 46B, 392 (1978).
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]

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