Quantum correlations in Newtonian space and time:arbitrarily fast communication or nonlocality

Nicolas Gisin(Dated: February 27, 2018)

We investigate possible explanations of quantum correlations that satisfy the principle ofcontinuity, which states that everything propagates gradually and continuously through spaceand time. In particular, following [J.D. Bancal et al, Nature Physics 2012], we show that anycombination of local common causes and direct causes satisfying this principle, i.e. propagating atany finite speed, leads to signalling. This is true even if the common and direct causes are allowedto propagate at a supraluminal-but-finite speed defined in a Newtonian-like privileged universalreference frame. Consequently, either there is supraluminal communication or the conclusion thatNature is nonlocal (i.e. discontinuous) is unavoidable.

It is an honor to dedicate this article to Yakir Aharonov, the master of quantum paradoxes.

I. INTRODUCTION

Correlations cry out for explanations [1]. This is truein all sciences, from correlations between measurementresults in quantum physics to correlations between earth-quakes and tsunamis in geophysics, and correlations be-tween tides and the moon’s positions in classical physics,to name but a few examples. Once a correlation has beenidentified, the next task of science consists in developing atheoretical model explaining the correlation. Such mod-els take the form of a story supported by mathematicalequations. Particularly challenging is the search of anexplanation for quantum correlations when consideringseveral measurements per party on two or more distantsystems initially in an entangled state.

In all sciences besides quantum physics, all correlationsare explained by a combination of only two basic mecha-nisms. Either a first system influences a second one, i.e.Direct Causation (DC), as for example the earthquakethat causes the tsunami. Or the correlated events sharea local Common Cause (CC) in their common past as tworeaders of this text whose readings are highly correlated.Sometimes the common or direct causes may be subtleand not easy to detect, as twins that look extraordinarilyalike thanks to common genes (local variables, i.e. CC),or as one’s yawning triggers others to yawn, thanks todelicate influences (i.e. DC).

Many correlations involve a combination of the twobasic mechanisms, common and direct causes, like forinstance the correlations between hockey players: theytrained together, hence share common causes, and, dur-ing games, influence each other.

Formally a correlation between two parties A and B isa conditional probability distribution p(a, b|x, y), wherea, b denote the measurement results collected by A and B,and x, y the measurement settings freely (i.e. indepen-dently from each other and from all CC and DC) cho-sen by A and B, respectively. This generalizes straight-forwardly to n parties. If A’s marginal p(a|x, y) ≡∑b p(a, b|x, y) depends explicitly on B’s choice y, then A

can get information about B’s choice by merely observ-

ing her local statistics. This is called signalling. The no-signalling principle states that A’s marginal is indepen-dent of B’s choice, p(a|x, y) ≡

∑b p(a, b|x, y) = p(a|x),

and B’s marginal is independent of A’s choice, p(b|x, y) =p(b|y). Note that all physical communication should becarried by some physical object (atoms, photons, energy,waves, etc). Hence, assuming only local Common Causescarried by the (localized) physical systems in Alice andBob’s hands, signalling would be non-physical commu-nication. But Direct Cause may allow signalling as dis-cussed in section(VIII).

This paper is organized as follows. In the next section,we present the intuition behind our result [2]. Next, insection III, we define formally v-causal models. Then,before presenting the main result in section V, we analyzethe case of DC (without additional variables) in sectionIV. Finally, we discuss experiments that could test ourresults in section VI and discuss the interpretation of ourresults.

II. EXPLANATIONS OF CORRELATIONS

First attempts at explaining correlations between dis-tant quantum measurement results assumed that thesource producing the entangled quantum systems pro-duces additional variables, hidden to today’s physics,which would locally (i.e. continuously) determine theprobabilities of the measurement results. This wouldprovide a local Common Cause explanation. Such localhidden variable models must obey the famous Bell in-equalities. But quantum theory predicts and experimentsconfirm that Bell inequalities can be violated; hence allexplanations based only on local common causes havebeen experimentally refuted1.

1 up to some combinations of loopholes that seem highly implau-sible; however, this being science, this logical possibility shouldbe addressed experimentally.

arX

iv:1

210.

7308

v2 [

quan

t-ph

] 2

1 Ja

n 20

13

2

Direct Cause explanations of quantum correlations re-ceived relatively little attention, compared to CC expla-nations (up to some noticeable exceptions, in particularEberhard who proposed an explicit model already in 1989[3]). This is due to the fact that Bell inequality violationshave been convincingly demonstrated between space-likeseparated measurements [4–6], hence a DC explanationswould require influences that propagate faster than light.

The assumption of faster than light influences does notrespect the spirit of relativity. However, the assumptionof a universal privileged reference frame with respect towhich a faster than light influence can be defined, is notin contradiction with relativity2. Think for example ofthe reference frame in which the micro-wave back groundradiation, residue of the big bang, is isotropic; our Earthpropagates with respect to this universal frame at thewell defined speed of 369 km/s in a direction known ateach moment [7].

There is thus no definite reason not to investigate thepossibility of explaining quantum correlation with a com-bination of DC and CC. Actually, many authors whothought seriously about quantum non-locality noticedthat correlation between distant events strongly suggestthat “something is going on behind the scene”, usingJohn Bell’s words [8, 9]. David Bohm and Basil Hiley,for example, have been very explicit when writing “it isquite possible that quantum nonlocal connections mightbe propagated, not at infinite speeds, but at speeds verymuch greater than that of light. In this case, we couldexpect observable deviations from the predictions of cur-rent quantum theory (e.g. by means of a kind of exten-sion of the Aspect-type experiment)” [10]. Let us alsonote that most (non relativistic) text books tell a storylike “first measurement collapses the entire wavefunction,hence changes (influences) the state of all systems entan-gled with the measured system”. Consequently, it is goodscientific practise to study the assumption that quan-tum correlations are caused by faster than light influ-ences propagating in a hypothetical universal privilegedreference frame and analyze its consequences.

We call v-causal all explanations that combine lo-cal Common Causes and Direct Causes where the in-fluence (describing the direct cause) propagates at asupraluminal-but-finite speed v defined in a hypotheticaluniversal privileged reference frame: c < v < ∞. Notethat such a universal privileged reference frame wouldbe quite similar to Newton’s space and time, but with agiven fixed maximal velocity v. It is thus quite familiarto physicists3, see Figure 1. When two events can be

2 One could also consider the history-fiction case that quantumtheory would have been developed before the discovery of rel-ativity. In such a case, quantum nonlocality would have beenequally surprising and fascinating and physicists would naturallyhave been led to search for explanations of these extraordinarycorrelations in terms of delicate influences yet to be discovered.

3 Though this strongly contrasts with ideas in quantum gravity

FIG. 1: Space-time diagram in the privileged reference frame.The shaded light cone is delimited by solid lines. Points insidethe v-cone (hatched), e.g. K2 and K3, are v-connected to K1;while points outside the v-cone, like K4, are not-v-connectedto K1. (Taken with permission from Nature [2]).

connected by a hidden influence at speed v, we say thatthey are v-connected; otherwise we say that the eventsare not-v-connected.

The kind of experiment that Bohm and Hiley had inmind to test such a DC or v-causal explanation is quiteintuitive: if the influence carrying the DC propagatesat finite speed, it should be possible to arrange an ex-periment between distant quantum systems with goodenough synchronization (in the universal privileged refer-ence frame), so that the influence doesn’t arrive on timeto establish the correlation. Thus, in such situations,the measured correlation should necessarily be local, i.e.satisfy all Bell inequalities, even in cases where quan-tum theory predicts a violation of some Bell inequality.Hence, v-causal explanations can’t reproduce all quan-tum predictions. Accordingly, they can be tested exper-imentally against quantum theory.

Such experiments face two intrinsic difficulties. First,since today we don’t know the hypothetical privilegedreference frame, it is not clear in which reference framethe synchronization should be optimized. Indeed, if twoevents are simultaneous in one frame, e.g. the labora-tory frame, then, according to special relativity, they arenot simultaneous with respect to others frames, e.g., tothe cosmic microwave background radiation frame. Sec-ond, within an assumed privileged reference frame, per-fect synchronization is impossible in practice; hence if

where space-time is sometimes thought of as an emergent con-cept, as e.g. in loop quantum gravity.

3

nonlocal correlations are observed, this only sets a lowerbound on the speed of the hypothetical hidden influence.Nevertheless, experiments have been carried out, settingstringent lower bounds of this speed, assuming the labframe [11], the microwave background radiation frame[12] and even scanning all possible privileged referenceframes [13, 14]. These experiments have excluded speedsup to about 50’000 times the speed of light.

At this point the case for a definite experimental testof DC explanation may seem quite hopeless: two-partyexperiments can only hope to increase the lower bound ofthe speed of the hypothetical hidden influence or to findthe breakdown of quantum theory. But in 2002 Vale-rio Scarani and myself noticed that the situation changesdramatically when analyzing situations with more thantwo parties [15, 16]. The original scenario we consideredinvolves 3 parties (see also Ryff [17] whose argument isrecalled in section IV). The general idea is the following.If two out of all parties measure simultaneously, e.g Boband Charlie are not-v-connected, then their correlationmust be local. If moreover, the correlations between theother pairs of parties, those whose measurements are v-connected, allow one to guarantee that Bob and Charlieshare nonlocal correlations, then one could infer a contra-diction with any v-causal explanation without the needfor any demanding synchronization. That one can inferthe nonlocality between Bob and Charlie without evermeasuring them in the same run of an experiment is quitecounterintuitive, though it is known that sometimes onecan infer a property of some quantum state or probabil-ity distribution from only the knowledge of some of theirmarginals [18, 19].

The next step was made by Stefan Wolf and col-leagues who introduced the concept of transitivity ofnonlocality [20]. They showed that, assuming only no-signalling, there are examples of 3-party correlations,p(a, b, c|x, y, z), such that if both marginals A-B and A-C are nonlocal, then the third marginal B-C is neces-sarily also nonlocal. This beautifully illustrates the ideaScarani and myself had in 2002. But unfortunately, Wolfand colleagues’s example uses correlations that can’t beachieved with measurements on quantum systems and,today, no quantum example of transitivity of nonlocal-ity has been found. This is why the example we presentin this paper doesn’t use the concept of transitivity ofnonlocality, but the theorem [2] recalled in section V.

To conclude this introduction let us consider some con-sequences of the assumption that v-causality is the ex-planation of all quantum correlations. As already men-tioned, this would imply that some predictions of quan-tum theory are wrong: if two events are not-v-connected,then their correlation would be local even in cases wherequantum theory predicts a violation of some Bell in-equality. But could this departure from quantum pre-dictions be used to communicate, in particular to com-municate faster than light? In the 2-party case, Aliceand Bob could arrange to be just at the border of beingv-connected. So, if Bob makes his measurement early

enough, the hidden influence doesn’t arrive on time andthey observe local correlations; but if Bob delays a lit-tle bit his measurement, then the influence arrives ontime and they observe quantum correlations. This, how-ever, can’t be used by Alice and Bob to communicate.Indeed, their local statistics would be identical in bothcases, whether the hidden influence arrives on time ornot; it is only later, once they compare their data, thatAlice and Bob can notice whether or not they violatedsome Bell inequality. Consequently, with only two par-ties, the hidden influence could remain hidden for ever:there would be a hidden layer at which faster than lighthidden influences carry Direct Causes and thus establishcorrelations that appear nonlocal, but at our higher levelnothing travels faster than light. In this paper, following[2], we prove that such a peaceful coexistence betweenrelativity and faster than light hidden influences can’texist. But for this we’ll need to consider more than twoparties.

III. v-CAUSALITY

In this section we define formally local Common Cause,Direct Cause and v-causal explanations. Readers whofeel they understand CC, DC and v-causality may like tojump to section IV.

Consider a 2-party scenario, denoted Alice and Bob,with measurement settings x and y and measurement re-sults a and b, respectively. The generalization to moreparties is straightforward, as summarized at the end ofthis section. The conditional probability distribution, orin short correlation, p(a, b|x, y), is the probability of re-sults a, b when the settings x, y are chosen.

A pure local Common Cause explanation ofp(a, b|x, y) assumes additional variables, traditionally la-beled λ, such that:

p(a, b|x, y) =∑λ

ρ(λ)p(a|x, λ)p(b|y, λ) (1)

where ρ(λ) denotes the probability that the additionalvariable assumes the value λ (note that λ may includethe quantum state ρ). For a justification see, e.g. [1, 21–24]. In a v-causal model, the information carried by thevariable λ propagates gradually and continuously fromsome common v-past of Alice and Bob. If v would be thespeed of light, this would merely be the usual intersectionbetween the past light cones. But here the common v-past is the intersection of wider, more open, cones, seeFig.1. Important in a common cause explanation is thatp(a|x, λ) doesn’t depend on y and symmetrically p(b|y, λ)is independent of x. Hence all correlations are due to thecommon local variable λ.

A pure Direct Cause explanation of p(a, b|x, y) as-sumes that there is an absolute time ordering of theevents at Alice and Bob (defined in the hypothetical uni-versal privileged reference frame). For example, assumeAlice is first to chose her measurement settings x and

4

collect her result a. Direct cause4 assumes that as soonas Alice performed her measurement, a signal - whichwe call a hidden influence - informs the rest of the uni-verse, in particular Bob, of her measurement setting xand result a. In this case there are two possibilities:

1. The information reaches Bob’s system before itproduces the result b, i.e. Alice and Bob are v-connected. In this case:

p(a, b|x, y, v-connected) = p(a|x)p(b|y, x, a) (2)

For example, quantum correlations between v-connected events can be described as due to DC:p(a|x) = Tr(AxaρA) where ρA Alice’s partial tracequantum state and Axa the projector representingher measurement, and p(b|y, x, a) = Tr(Byb ρ

xa)

where ρxa =Ax

aρAxa

Tr(Axaρ)

is Bob’s reduced state that

depends on Alice’s measurement setting x and re-sult a. Note that in this case direct cause exactlyreproduces the quantum prediction: p(a, b|x, y, v-connected) = Tr(Axa ⊗B

yb · ρ).

2. Bob’s system has to produce the result b before theinformation carried by the hidden influences arrivesfrom Alice’s system, i.e. Alice and Bob are not-v-connected:

p(a, b|x, y, not-v-connected)

= p(a|x)p(b|y) (3)

In the case that Bob’s probability depends only onhis local quantum state ρB = TrA(ρ), one has:

p(a, b|x, y, not-v-connected)

= Tr(Axa · ρA) · Tr(Byb · ρB) (4)

In general, for entangled states ρ, this predictiondiffers from the quantum prediction.

A v-causal explanation of p(a, b|x, y) combines addi-tional local variables and hidden influences5, all propa-gating at a speed v (or lower) defined in the universalprivileged reference frame. This frame defines an abso-lute time ordering, as for direct cause explanations. Hereagain one has to distinguish two possibilities dependingon whether Alice and Bob are v-connected or not:

4 Standard text book descriptions of measurements collapsing thequantum state is an explicit example of a hidden influences ex-planation; however, in such descriptions the influence propagatesat infinite speed. Hence it is more a direct action at a distancethan an influence propagating in space and time. Note that be-cause of the infinite speed, all parties are v-connected. Suchdescriptions also require a universal privileged reference frame.

5 The De-Broglie-Bohm pilot wave model is an explicit example ofa v-causal explanation; however, in this model the influence prop-agates at infinite speed. Hence it is more a direct action at a dis-tance than an influence propagating in space and time. Note thatbecause of the infinite speed, all parties are v-connected, henceBohm’s model recovers all quantum predictions. This model alsorequires a universal privileged reference frame.

1. The information reaches Bob’s system before itproduces the result b, i.e. Alice and Bob are v-connected. In this case:

p(a, b|x, y, v-connected)

=∑λ

ρ(λ)p(a|xλ)p(b|y, λ, x, a) (5)

Since we look for an explanation of quantum corre-lations, one expects that, in the case of v-connectedevents, quantum correlations are reproduced.

2. Bob’s system has to produce the result b before theinformation carried by the hidden influences arrivesfrom Alice’s system, i.e. Alice and Bob are not-v-connected:

p(a, b|x, y, not-v-connected)

=∑λ

ρ(λ)p(a|x, λ)p(b|y, λ) (6)

where λ includes the quantum state ρ. Conse-quently, in any v-causal model, unconnected eventsmust satisfy all Bell inequalities.

The generalization to an arbitrary number of partiesshould be straightforward: when a system undergoes ameasurement it takes into account all the informationit received, whether additional local variables or hiddeninfluences, and sends out information about itself in alldirections by hidden influences propagating at speed v.Since we are looking for an explanation of quantum corre-lations, one expects that, whenever possible, v-connectedevents reproduce quantum correlations. However, un-connected events necessarily produce local correlations,hence correlations that may differ from the quantum pre-dictions. This constraint is what limits the power of v-causal explanations and makes experimental tests possi-ble.

Note that the speed of light c doesn’t appear in thedefinitions of local Common Cause and Direct Cause,nor v-causality 6.

IV. NO DIRECT CAUSE EXPLANATION

In this section we study the assumption that correla-tions between quantum measurement results are due toDC carried by hidden influences propagating at a finitebut supraluminal speed v. More precisely, we considerhidden influence plus the usual quantum state, but no

6 nor does c appear in the definition of ”Bell locality” (1). Nev-ertheless, physicists have always been interested in tests of Bellinequalities between space-like separated events, i.e. betweenevents not-c-connected. This illustrates that v-causal modelswere always in the back of the mind of those physicists, thoughwith v = c.

5

FIG. 2: Spacial configuration of the 3-party scenario discussedin section IV to show that pure Direct Cause leads to sig-nalling.

additional local variables. This section is greatly inspiredby [17] (note that Eberhard published a related argumentalso involving 3 parties [3]).

Consider a 3-party scenario, Alice, Bob and Charlie,where Alice is far away from both Bob and Charlie. Boband Charlie are relatively close to each other, but distantenough (in the hypothetical universal reference frame)so that they can synchronize their measurements wellenough to be not-v-connected, see Fig. 2. Alice, Bob andCharlie know the relative positions of each other and atwhat time Bob and Charlie perform their measurements.Assume they share a GHZ state Ψ = |0, 0, 0〉+|1, 1, 1〉 andall measure σz. Quantum theory predicts that all threecollect the same result: a = b = c. We shall see thatif this correlation is due to some supraluminal hiddeninfluence (without additional variables), then Alice couldcommunicate faster than light to Bob and Charlie (Boband Charlie need to collaborate).

The argument runs as follows. First, if Alice chooses tocommunicate “yes”, she performs her measurement earlyenough that the hidden influence arrives on time to Boband Charlie. In this case the hidden influence tells Boband Charlie’s system which result a Alice obtained, henceBob and Charlie’s system produce that same result: b =a and c = a. Next, if Alice chooses to communicate “no”,she doesn’t perform any measurement, or only too latefor the hidden influence to arrive on time. In this caseBob and Charlie obtain random and independent results(recall that they are not-v-connected, hence their resultare produced independently of each other), whence halfthe time b 6= c. Consequently, once Bob and Charliecompared their results (which they can do in a time veryshort relative to the time light would take to propagatefrom Alice to them), they can infer with good probabilityAlice’s message.

This is faster than light communication from Alice toBob-Charlie. By elongating the triangle the speed of thiscommunication gets arbitrarily close to the speed v of thehidden influence. Hence, the hidden influence doesn’tremain hidden, but can be activated.

This simple example shows that with 3 parties one canactivate the hidden influence, something impossible withonly 2 parties. However, this example also shows thatthere is a simple way around the argument. Indeed, thecorrelation is a simple and local one: a = b = c. Hence,one could merely supplement the DC explanation witha shared random bit r and assume that in the case thehidden influence doesn’t arrive on time, all systems pro-

tpriv.

xprivileged•Alice Bob Charlie Dave

•

• •

A

B C

D

⟨⟨⟨⟨ABD⟩⟩⟩⟩ is quantum

⟨⟨⟨⟨ACD⟩⟩⟩⟩ is quantum

⟨⟨⟨⟨BC⟩⟩⟩⟩ is localeven ifconditionedon A and D

FIG. 3: Space-time configuration in the privileged referenceframe of the 4-party scenario discussed in section V to showthat all v-causal models lead to signalling. (Taken with per-mission from Nature from [2]).

duce the result r. This motivates the investigation ofv-causality, where DC is combined with additional localvariables as explained in section III and analyzed in thenext section.

V. NO v-CAUSAL EXPLANATION

At this stage of the search for an explanation of quan-tum correlations, local common causes and hidden in-fluences are both individually excluded. The first onepredicts Bell inequalities that have been violated, whilethe second one can’t remain hidden as recalled in the pre-vious section. Let us thus analyse the hypotheses thatv-causality, i.e. an arbitrary combination of Direct andCommon Causes, is the explanation of all correlations.This might sound bizarre. But quantum correlations arebizarre and there is simply no other type of explanationsthat satisfy the principle of continuity (we discuss thisprinciple in more detail in sectionVII). This section isgreatly inspired by [2].

Consider the 4-party configuration of figure 3, repre-sented in the hypothetical privileged reference frame. Al-ice, Bob, Charlie and Dave have a choice between twomeasurement settings, labeled x, y, z and w and collectbinary results a, b, c, d ∈ {−1,+1}, respectively. Alicemeasures first, hence is not influenced by any of the otherparties. Next, Dave measures at a time such that the hy-pothetical influence from Alice arrives on time to Dave.Finally, Bob and Charlie measure quasi-simultaneously,i.e. Bob and Charlie are not-v-connected, but such thatthe hypothetical influences from Alice and Dave arriveon time both to Bob and to Charlie.

If we were considering only DC, the joined probability

6

would read:

p (a, b, c, d|x, y, z, w) = (7)

p (a|x) · p(d|w, x, a) · p(b|y, x, a, w, d) · p(c|z, x, a, w, d)

It is not difficult to see that the correlation (7) leads tosignalling from Alice to Bob-Charlie-Dave (who need tocooperate). But in this configuration, contrary to thetriangular configuration of the previous section, we canexclude the possibility that additional variables allow oneto avoid the activation of the hypothetical hidden influ-ence.

The idea is to find an inequality satisfied by all no-signalling correlations where the not-v-connected partiesare local with the following two properties:

1. all terms in the inequality involve only v-connectedparties (hence, to evaluate the inequality one neverhas to measure in a same round of the experimentnot-v-connected parties, one thus avoids the syn-chronization difficulty),

2. the n-party correlation can be violated by quan-tum correlations (i.e. quantum theory predicts aviolation of the inequality).

The technical difficulty of this strategy is that, firstone has to study the intersection of the n-party no-signalling polytope with the local potytope of the not-v-connected parties. Next, one has to project this intersec-tion polytope on the subspace of correlations containingonly terms corresponding to v-connected parties.

This strategy can obviously not work with only twoparties (both would either be v-connected or bothnot-v-connected). Hence, with my co-authors of [2] wespent a long time searching for an example involving 3parties, one pair being not-v-connected and two pairsv-connected. But no example has been found, thoughthe search continues, varying the number of inputs(measurements settings) and outcome for each party[25]. The breakthrough came when Jean-Daniel Bancaland Stefano Pironio had the courage to consider 4parties in the configuration of figure 3. After heavynumerical search they found the following [2].

Theorem Let p(a, b, c, d|x, y, z, w) be a correlation, i.e.a conditional probability distribution, with binary inputsx, y, z, w ∈ {0, 1} and outcomes a, b, c, d ∈ {−1,+1}.If

1. The correlation p(a, b, c, d|x, y, z, w) is non-signalling, and

2. p(b, c|y, z, a, x, d, w) is local7 for all a, x, d, w,

7 i.e. satisfy the Clauser-Horn inequality: p(b = c =0|0, 0, a, x, d, w) + p(b = c = 0|0, 1, a, x, d, w) + p(b = c =0|1, 0, a, x, d, w) − p(b = c = 0|1, 1, a, x, d, w) − p(b = 0|y =0, a, x, d, w)− p(c = 0|z = 0, a, x, d, w) ≤ 0 and all its symmet-ric forms obtained by permuting the inputs and outcomes.

then S ≤ 7, where

S = −3〈A0〉 − 〈B0〉 − 〈B1〉 − 〈C0〉 − 3〈D0〉− 〈A1B0〉 − 〈A1B1〉+ 〈A0C0〉+ 2〈A1C0〉+ 〈A0D0〉+ 〈B0D1〉− 〈B1D1〉 − 〈C0D0〉 − 2〈C1D1〉+ 〈A0B0D0〉+ 〈A0B0D1〉+ 〈A0B1D0〉− 〈A0B1D1〉 − 〈A1B0D0〉 − 〈A1B1D0〉+ 〈A0C0D0〉+ 2〈A1C0D0〉 − 2〈A0C1D1〉 (8)

In (8) 〈A1B0〉 denotes the average of the product ofAlice and Bob’s outcomes when Alice chooses x = 1 andBob y = 0 and similarly for the other terms.

The above inequality S is remarkable because none ofits 23 terms involves both Bob and Charlies, hence it canbe evaluated without ever measuring Bob and Charlie inthe same run of an experiment. Nevertheless,

1. Assuming no-signalling, a violation implies thatBob and Charlie share nonlocal correlations, i.e.correlations that can’t be explained by CommonCauses, and

2. Assuming that Bob and Charlie are local, as theyare in any v-causal model, a violation implies thatp(a, b, c, d|x, y, z, w) is signalling.

It is not difficult to check that the inequality S ≤ 7can be violated by the following 4 qubit state [2]

|Ψ〉 =17

60|0000〉+

1

3|0011〉 − 1√

8|0101〉+

1

10|0110〉

+1

4|1000〉 − 1

2|1011〉 − 1

3|1101〉+

1

2|1110〉 (9)

with the measurements

A0 = −UσxU† A1 = UσzU† (10)

B0 = H B1 = −σxHσx (11)

C0 = −D0 = σz C1 = D1 = −σx (12)

where U = cos( 4π5 )σz − sin( 4π

5 )σx, the σ’s denote thePauli matrices and H the Hadamard matrix. Quantumtheory predicts for these state and measurement settingsS ≈ 7.2.

Accordingly, the supraluminal hidden influence in anyv-causal model can be activated. Indeed, in any v-causalmodel Bob and Charlie are local, hence, one can deducefrom the quantum prediction that the 4-party correlationis signalling (recall that the 4-party correlation is notquantum, because Bob and Charlie are not-v-connected,only the 3-party marginals A-B-D and A-C-D are quan-tum, but this suffices to evaluate S).

Consequently, at least one of the four 3-partymarginals depends on the fourth’s input. Consider first

7

FIG. 4: In the 4-party scenario, signalling leads to faster thanlight communication. Here we illustrate the case where thesignalling goes from A to BCD; the case D→ABC is similar(the other cases don’t happen, because they are quantum, seetext). (Taken with permission from Nature from [2]).

the A-B-D 3-party marginal; since A-B-D are all v-connected, p(a, b, d|x, y, w) is quantum and thus non-signalling (it doesn’t depend on Charlie’s input z). More-over, the A-B-D correlation can’t depend on Charlie’s in-put z, because z is chosen outside of A-B-D past v-cones.Similarly for the A-C-D 3-party marginal. Consequently,it must be either A-B-C that depends on Dave’s inputw or B-C-D that depends on Alice’s input x (or both).Both cases are similar; let us thus consider the case thatp(b, c, d|y, z, w, x) depends explicitly on x. This is sig-nalling from Alice to Bob-Charlie-Dave. Moreover, thiscan be used for faster than light communication: it suf-fices that Bob and Charlie send (at the speed of light)their inputs y, z and outcomes b, c to Dave so that Davecan evaluate their 3-party marginal B-C-D. Since thismarginal depends on Alice’s measurement setting choicex, Alice can communicate to Dave. Figure 4 shows thatthis communication can be faster than light. By movingB-C-D away from A, but such that the hidden influencefrom Alice still arrives on time to all of them, one canmake this faster than light communication tend to thespeed v of the hidden influence.

In summary, the hidden influence of any v-causal ex-planation of quantum correlation can never remain hid-den: it necessarily allows for faster than light communi-cation. We’ll come back to this remarkable conclusion insection VIII.

VI. EXPERIMENT

In this section we consider how experiments could testthe contradiction we have established between quantumtheory, v-causality and no faster than light communica-tion. At first, one may wonder whether such an exper-

iment is necessary at all. Indeed, quantum correlationshave been measured abundantly. Hence, it seems highlylikely that the state (9) and the quantum measurements(10)-(12) can be realized with good enough approxima-tion to violate inequality (8). Moreover, the very as-sumption that quantum correlations are explained by v-causality implies that the ABD and the ACD correla-tion predicted by any v-causal model are identical to thequantum prediction, hence that any v-causal model vio-lates the inequality. If not, the v-causal model would notbe an explanation for the quantum correlation8. Further-more, if it would turn out impossible to violate inequality(8), then quantum theory would fail even in cases werethe events are v-connected. This would be very difficultto explain and v-causality might not be of much help.This is in sharp contrast to Bell’s inequality: had it turnout impossible to violate Bell’s inequality, local CC wouldhave been vindicated. Hence, whether or not one even-tually observes a violation of the inequality (8), in bothcases explanations based on v-causality seem difficult tomaintain!

But, physics being an experimental science, one shouldcheck that correlations violating the inequality we used inthe previous section to derive our conclusion can indeedbe realized.

So, imagine a source producing a state close to the4 qubit state (9) and distributing each qubit to Alice,Bob, Charlie and Dave. Alice is first to choose her mea-surement setting x, measure her qubit and collect heroutcome a. Alice and her qubit may be aware of thelocations of her partners, who may perform some mea-surement, or are measuring quasi simultaneously, suchthat the corresponding hidden influence did not reach heryet. However, Alice can’t know when such possible mea-surements will be performed by her partners, or whetherthey will be performed at all (the so-called “free-will” as-sumption). Accordingly, Alice (more precisely her qubit)has to send out her hidden influence at speed v indepen-dently of when Bob, Charlie and Dave may measure (ornot measure) their qubits.

Dave should be second to measure, but not too early,so as to make sure that Alice’s hidden influence reacheshim on time. This can be guaranteed by merely lettingDave measure his qubit at a time such that even lightwould arrive on time. Since the hidden influence propa-gates faster than light, it will necessarily also arrive ontime, irrespectively of which reference frame is the privi-leged one. Here we assume that Alice’s hidden influencealways propagates at the same speed v, independentlyof the protocol of the experiment. This is a standardassumption in science: one never assumes that the ex-perimental protocol changes the laws one is testing. Insummary, it is easy to guarantee that Dave measures far

8 Though, if quantum theory is falsified, then one would no longerbe looking for an explanations of all quantum correlations.

8

enough in the future to respect the time ordering of figure3.

This just leaves Bob and Charlie. They should bothmeasure in the future of Dave so that its hidden influencearrives on time. This can again be achieved by settingBob and Charlie in the future light cone of Dave (andthus also of Alice). However, according to the configu-ration depicted in figure 3, Bob and Charlie should bewell enough synchronized to guarantee that no hiddeninfluence from one can reach the other. This is impos-sible without knowing an upper bound on the speed ofthe hidden influence and the privileged frame. This dif-ficulty is circumvented, as already explained in the pre-vous section, by the observation that in the inequality(8), no term involves both Bob and Charlie. Hence, onedoesn’t need to ever measure them in a same round ofthe experiment. It suffices that, after Dave measured hisqubit, a random choice is made by the experimentalistto measure either Bob or Charlie’s qubit. In each case,another, fourth (independent) random choice is made toselect the measurement setting. Both these choices aremade in the future light cone of Dave. Again, we assumethat the qubit chosen to be measured, whether it is Bob’sor Charlie’s, produces a result that is independent of theprotocol. In other words, in case Bob’s qubit is chosen tobe measured, the probability of the result b is the sameas if Charlie’s qubit would be measured simultaneously:Bob’s result probability can’t depend on when Charlie’squbit is measured as long as Charlie qubit can’t influ-ence Bob’s. And if Charlie’s qubit is not measured at all,Bob’s result probability can’t depend on “when Charlie’squbit is not measured”.

In summary, a first random bit decides Alice’s mea-surement setting x, next in the absolute future a secondrandom bit chooses Dave’s setting w, finally, again inthe absolute future, a third random bit decides whetherBob’s or Charlie’s qubit is measured and a fourth ran-dom bit decides the measurement setting y or z. Notethat all these 4 random bits must be independent of thehypothetical additional variables and hidden influences,as in Bell inequality analysis (this is sometimes called the“free will” or the “measurement independence” assump-tion [26, 27]).

In this way, the experimental test of quantum predic-tions for the configuration depicted in figure 3 can berealized. If a violation is observed, as one expects fromquantum theory, then one has to conclude that

1. either the hypothetical hidden influence can’t re-main hidden, but necessarily leads to signalling andto faster than light communication,

2. or, all v-causal explanations are ruled out, i.e. nocombination of Direct Cause and local CommonCause can explain the experimental result.

Both these alternatives are fascinating and will be dis-cussed in the conclusion section.

One might be surprised that the proposed experimentdoesn’t involve any space-like separated measurements.

But, as mentioned at the end of section III, the speedof light doesn’t appear in the definition of v-causality.Hence, according to v-causality, one doesn’t expect anydifference when measurements are time-like or space-like separated. Furthermore, signalling between time-likeseparated events would be about as bizarre as betweenspace-like separated events. Indeed, imagine that Aliceis located in a safe, e.g. in the basement of the Swiss na-tional bank. One expects that this would not affect thecorrelations between her measurements and those of herpartners, wherever they are located. In particular theycould be in the future light-cone of Alice, somewhere out-side of the bank. But then, signalling from Alice to BCD,as v-causality and the violation of (8) predict, impliesthat Alice could communicate to her partners, whateverphysical security measures and isolation one imposes onAlice9!

VII. NEWTON AND THE PRINCIPLE OFCONTINUITY

It is not the first time in history that physics is con-fronted with nonlocality. Actually, physics almost al-ways presented a nonlocal world-view of nature, first withNewton’s theory of universal gravitation, then with quan-tum nonlocality. Only during short time window of about10 years did physics present a local world-view.

Newton was very concerned by the nonlocal predic-tions of his theory of universal gravitation. Indeed, henoticed that his theory predicts that any change in thelocal configuration of matter would have an immediateeffect on the entire universe. Hence, by moving to theleft or to the right a stone on the moon10, one could, inprinciple, signal at arbitrary speed to Earth and to anyplace in the universe. Let us read how the great mandescribed the situation: [28]:

That Gravity should be innate, inherent and essentialto Matter, so that one Body may act upon another at aDistance thro a Vacuum, without the mediation of anything else, by and through which their Action and Forcemay be conveyed from one to another, is to me so great anAbsurdity, that I believe no Man who has in philosophi-cal Matters a competent Faculty of thinking, can ever fallinto it. Gravity must be caused by an Agent acting con-stantly according to certain Laws, but whether this Agentbe material or immaterial, I have left to the Considera-tion of my Readers.

Accordingly, “no action at a distance” is not a princi-ple of relativity nor of Einstein, but is part of Newtonian

9 This would be similar to signalling using gravitation - no way toprevent it - but at the speed v.

10 To move the stone one shouldn’t take support on the moon, asthis would not move the center of mass of the moon-&-stone, butuse a small rocket.

9

space-time. Let us emphasize that “no action at a dis-tance” implies that nothing propagates at infinite speed,in particular there are no infinite speed influences nor∞-causality.

Usually, quantum correlations are seen as being in ten-sion with (special) relativity, remember Shimony’s state-ment about the peaceful coexistence of quantum theoryand relativity. But it is natural to go beyond these ten-sions and investigate the consequences of assuming thatthe correct interpretation of Lorentz transformation isnot mere geometry of space-time, but real Fitzgerald con-tractions of lengths and Larmor dilation of time intervals,as Lorentz and Poincare themselves thought and as JohnBell considered [9, 29]. Hence, the interest for studyingquantum correlations in Newtonian space-time, or, equiv-alently for that matter, in a universal privileged referenceframe.

Notice that all v-causal explanations of correlationssatisfy a principle of continuity that states that every-thing (mass, energy and information) propagates gradu-ally and continuously through space as time passes, i.e.nothing jumps instantaneously from here to there. Inother worlds, there is no action at a distance. Recip-rocally, all explanations of correlations that satisfy theprinciple of continuity are v-causal. Hence, Newton andEinstein would have bet on a v-causal explanation of allcorrelations, including quantum correlations.

An experimental violation of the inequality S ≤ 7 ei-ther implies a violation of the principle of continuity orimplies faster than light communication.

VIII. NO-SIGNALLING IN v-CAUSALITY

No-signalling is generally considered as a fundamentalprinciple that has to hold in any meaningful physical the-ory. However, if the correlations between some events aredue to hidden influences, then there is no reason to as-sume that the influences don’t allow one to signal (at thespeed of the hidden influence or slower). This is for exam-ple the case with gravity. Had someone before Einsteinhad the technology to check the correlation between thedisplacement of a stone on the moon and the weight ofsome mass on Earth, even when displaced and measuredsimultaneously, he would have observed a null correla-tion (at least for good enough synchronization) and thushave falsified Newton’s theory of universal gravitation.He could also have observed that the correlation estab-lishes when the weight measurement is performed about asecond after the displacement of the mass on the moon.This would have allowed him to signal at the speed ofwhat was then a hidden influence, i.e. the speed of gravi-tons that, according to general relativity, carry the causeof the change in the weight of the mass on Earth. This isa typical Direct Cause explanation. Note that one couldhave used this hidden influence to signal even withoutknowing the theory of general relativity.

Similarly, if the speed of the hidden influence that ex-

plains quantum correlations propagates faster than thespeed of light, then the corresponding signalling wouldequally be faster than the speed of light. Consequently,there are only two possibilities:

1. either the hidden influence remains hidden for ever,i.e. is intrinsically hidden11, hence doesn’t allow forsignalling, or

2. the hidden influence can be used to communicate ata speed equal or lower than the speed of the hiddeninfluence, i.e. the hidden influence doesn’t remainhidden.

In section V we demonstrated that the hypothetical hid-den influence of all v-causal model can’t remain hidden,but on the contrary leads to faster than light communi-cation at the level of the classical measurement settingsand results. Hence, the first of the above two alternativeis excluded.

IX. CONCLUSION

The main conclusion of this paper is that an experi-mental violation of the inequality S ≤ 7 would imply

1. either a violation of the principle of continuity (thatstates that everything propagates gradually andcontinuously through space as time passes as dis-cussed, in section VII), i.e. the falsification of allv-causal models, or

2. the possibility of faster than light communicationat the level of the classical measurement settingsand results.

It is unlikely that many physicists will contemplate se-riously the second alternative12. However, one shouldrealize that the first alternative is about as difficult toswallow as the second one. A violation of the principle

11 I am quite suspicious of explanations relying on intrinsically hid-den stuff, hence I dislike this part of the alternative.

12 One recent exception is B. Cocciaro [30]. In this paper the authoralso recalls that faster than light communication in one univer-sal global privileged reference frame, as consider in this paper,doesn’t lead to the “grand father” time paradox. Indeed, fortime paradoxes one should communicate to one’s own past; thisrequires a go-&-return communication. But if both the go andthe return signal are defined in the same reference frame and atthe same - possibly supraluminal - speed v, then the “return”signal will necessarily arrive in the absolute future of the startof the “go” signal. It is straightforward to see this in the priv-ileged frame. But then, the start of the “go” and the arrival ofthe “return” signals are necessarily also time-like in all other ref-erence frames, hence the impossibility to communicate to one’sown past. This is not new and was emphasized, e.g., in [22, 30–32]. Consequently, supraluminal communication might not havesaid it’s last word.

10

of continuity implies that the world is truly and defini-tively not local, i.e. Nature is not continuous, but nonlo-cal. This conclusion has already been claimed by manyphysicists (including this author), though only based onthe violation of Bell inequality between space-like sep-arated events. These physicists made the (admittedlyhighly plausible) assumption that space-time is describedby relativity. In this paper we have extended the conclu-sion: even if one is willing to consider a Newtonian-typeprivileged reference frame, but without faster than lightcommunication, the conclusion that Nature is nonlocal isunavoidable.

Should then Physicists give up the great Enterprize ofexplaining how Nature does it [33]? Certainly not! Butphysicists have only two options:

1. Pursue the search for the speed of v-causal explana-tions by improving the ”Salart-type” experiments[13, 14]. Note that the finding of such a speed wouldfalsify both quantum theory and relativity, a resultnot many physicists are willing to envisage. How-ever, the tension between these two pillars of to-day’s physics may well dissolve not merely by sav-

ing one of them at the cost of the other, but byfinding the limits of both theories. Accordingly,”Salart-type” experiments are still needed, but theresult of [2] – recalled in this paper – shows thata positive result would definitively lead to fasterthan light communication, hence it would not onlyfalsify quantum theory but also falsify relativity.

2. Accept quantum nonlocality and enlarge our storytool-box by inventing new tools – necessarily non-local – to tell explanatory stories. Possibly some-thing like “one random event can manifest itself atseveral locations”.

Acknowledgment

This article greatly profited from numerous exchanges withmy co-authors of [2] and from comment by Rob Thew andmany colleagues over the years. This work has been sup-ported by the ERC-AG grant QORE, the CHIST-ERA DIQIPproject, and by the Swiss NCCR Quantum Science and Tech-nology - QSIT.

[1] J. S. Bell, Speakable and Unspeakable in Quantum Me-chanics: Collected papers on quantum philosophy (Cam-bridge University Press, Cambridge, 2004).

[2] J.D. Bancal, S. Pironio, A.Acin, Y.C. Liang, V. Scaraniand N. Gisin, arXiv:1110.3795, Nature Physics, in press(2012).

[3] Ph. H. Eberhard, A realistic model for Quantum Theorywith a locality property, in: Quantum theory and picturesof reality, W. Schommers (ed.) New York: Springer 1989,p. 169-215.

[4] A. Aspect, J. Dalibard, G. Roger, Phys. Rev. Lett. 49,1804 (1982).

[5] W. Tittel, J. Brendel, N. Gisin, and H. Zbinden, Phys.Rev. Lett. 81, 3563-3566 (1998).

[6] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, andA. Zeilinger, Phys. Rev. Lett. 81, 5039 (1998).

[7] C. Lineweaver et al., Astrophys. J. 470 (1996) 38.http://pdg.lbl.gov

[8] J. S. Bell, La nouvelle cuisine, in reference [1].[9] J.S. Bell in The Ghost in the Atom, eds P.C.W Davies

and J.R. Brown, Cambridge University Press, pp 45-57(1993).

[10] D. Bohm and B.J. Hiley, The Undivided Universe, Rout-ledge, London and NY 1993 (p. 347 of the paperbackedition).

[11] N. Gisin, V. Scarani, W. Tittel and H. Zbinden, “ 100years of Q theory ”, Proceedings, Annal. Phys. 9, 831-842 (2000); quant-ph/0009055; Andre Stefanov, HugoZbinden and Nicolas Gisin, Physical Review Letters 88,120404 (2002); N. Gisin, Sundays in a Quantum Engi-neer’s Life, Proceedings of the Conference in Commem-oration of John S. Bell, Vienna 10-14 November 2000.

[12] V. Scarani, W. Tittel, H. Zbinden and N. Gisin, Phys.Lett. A 276, 1-7 (2000)

[13] D. Salart, A. Baas, C. Branciard, N. Gisin, and

H. Zbinden, Nature 454, 861 (2008).[14] B. Cocciaro, S. Faetti and L. Fronzoni, Phys. Lett. A

375, 379 (2011).[15] V. Scarani and N. Gisin, Phys. Lett. A 295, 167 (2002)[16] V. Scarani and N. Gisin, Braz. J. Phys. 35, 2A (2005).[17] L.C. Ryff, arXiv:0903.1076[18] N. Linden, S. Popescu, and W. K. Wootters, Phys. Rev.

Lett. 89, 207901 (2002); N. Linden and W. K. Wootters,Phys. Rev. Lett. 89, 277906 (2002).

[19] L.E. Wurflinger, J.D. Bancal, A. Acin, N. Gisin and T.Vertesi, Phys. Rev. A 86, 032117 (2012).

[20] S. Coretti, E. Hanggi, and S. Wolf, Phys. Rev. Lett. 107,100402 (2011).

[21] D. Mermin, Am. J. Phys. 49, 940 (1981).[22] T. Mauldin, Quantum Non-Locality and Relativity:

Metaphysical Intimations of Modern Physics (BlackwellPublishers, Oxford, 2002).

[23] T. Norsen, John S. Bell’s concept of local causality, Am.J. Phys. 79, 1261-1275 (2012).

[24] N. Gisin, Found. Phys. 42, 80-85 (2012).[25] T. Barnea, master thesis, University of Geneva, 2012.[26] M. J. W. Hall, Phys. Rev. Lett. 105, 250404 (2010).[27] J. Barrett and N. Gisin, Phys. Rev. Lett. 106, 100406

(2011).[28] Isaac Newton, Papers & Letters on Natural Philosophy

and related documents, page 302, Edited, with a gen-eral introduction, by Bernard Cohen, assisted by RobertE. Schofield Harvard University Press, Cambridge, Mas-sachusetts, 1958

[29] J.S. Bell, How to teach special relativity, in [1][30] B. Cocciaro, arXiv:1209.3685.[31] F. Reuse, Ann. Phys. 154, 161 (1984).[32] P. Caban, J. Rembielinski, Phys. Rev. A 59, 4187 (1999).[33] N. Gisin, ”Quantum Nonlocality: How Does Nature Do

It?”, Science, 326:1357-1358, 2009.