Quant-Ph/0512168V1 20 Dec 2005 Uly M Fcus,Ntwiigfrenti,Btfor Ac- Sub- the Well, but Somewhat of Einstein, (Necessarily [2])

Home , Nicolas Gisin

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PACS local with strictly ”nonlocality of only concept the very relativity, the of that spirit note We gap. years mhsz htpyishsawy ie olcldescript nonlocal a given always has physics that emphasize ratruh vrtels e er.Frtefis iei is it time first the For years. few last the over breakthroughs signaling” rmNwoinnnoaiyt unu olclt n beyo and nonlocality quantum to nonlocality Newtonian From erve h oghsoyo olclt npyiswt spe with physics in nonlocality of history long the review We .INTRODUCTION I. ru fApidPyis nvriyo eea 21Geneva 1211 Geneva, of University Physics, Applied of Group immediately NEWTON rmteoutside the from ecflco-existence peaceful immediate a eaiiyb osdrdcmlt ? complete considered be relativity Can oie.Wa trou- What modified. hti ihu l h unu hsc ibr pc artil space Hilbert physics quantum the all without is that , ffc,ie the i.e. effect, 3 between [3] Dtd coe 1 2018) 31, October (Dated: ioa Gisin Nicolas nyte sorwih ffce.Ti s eiv,the believe, I is, This mechanical greatest engineer the affected. 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Well, subjective somewhat engineer! a good is a this mu and physicist reader, nega- a theorist no me, For has good ”engineer” a opposite. mouth the my quite in connotation, that tive well know friends My eea eaiiycnas ese samechanical a as seen be also can relativity General the- new radically three introduced Einstein 1905 In o fNtr,ecp uigasot10 short a during except Nature, of ion 1 osbet td nnoaiywithout ”nonlocality study to possible I.ENTI,TEGREATEST THE EINSTEIN, III. u inln”i oal oeg othe to foreign totally is signaling” out faltimes: all of EHNCLENGINEER MECHANICAL ilepai nteconceptual the on emphasis cial ,Switzerland 4, isentre hsc into physics turned Einstein ey We lery. nd informed tbe st and 2 a local theory! was, during many decades, not considered as serious. But this was indeed the real state of affairs: ask any older professors, a vast majority of them still believes that it IV. QUANTUM MECHANICS IS NOT is unimportant. Let me add two little stories that illus- MECHANICAL trate what the situation was like. John Bell, the famous John Bell of the Bell inequalities and of the Bell states, Only about ten years after general relativity came never had any quantum physics student. When a young quantum mechanics. This was quite an extraordinary physicist would approach him and talk about nonlocal- revolution. Until then, greatly thanks to Newton and ity, John’s first question was: ”Do you have a permanent Einstein’s genius, Nature was seen as made out of position?”. Indeed, without such a permanent position it many little billiard-balls that mechanically bang into each was unwise to dare talking about nonlocality! Notice that other. Yet, quantum mechanics is characterized by the John Bell almost never published any of his remarkable very fact that it no longer gives a mechanical description and nowadays famous papers [7] in serious journals: the of Nature. The terminology quantum mechanics is just a battle with referees were too ... time wasting (not to use a historical mistake, it should be called Quantum Physics more direct terminology). Further, if you went to CERN as it is a radically new sort of physical description of where John Bell held a permanent position in the theory Nature. department and asked at random about John’s contribu- But this new description let nonlocality back into tions to physics, his work on the foundation of quantum physics would barely be mentioned (true enough, he had Physics! And this was unacceptable for Einstein. 2 It is remarkable and little noticed that since Newton, so many other great contributions!) . physics gave a local description of Nature only during some 10 years, between about 1915 and 1925. All the Anyway, so quantum nonlocality remained for decades rest of the time, it was nonlocal, though, with quan- in the curiosity lab and no one paid much attention. But tum physics, in quite a different sense as with Newton in the 1990’s two things changed. First, a conceptual gravitation. Indeed, the latter implies the possibility of breakthrough happened thanks to Artur Ekert and to arbitrarily fast signaling, while the former prohibits it. his adviser David Deutsch [9]. They showed that quantum nonlocality could be exploited to establish a cryptographic key between two distant partners and that the V. NON-LOCALITY ACCORDING TO EINSTEIN confidentiality of the key could be tested by means of Bell’s inequality. What a revolution! This is the first time that someone suggested that quantum nonlocality In 1935 two celebrated papers appeared in respectable is not only real, but that it could even be of some use. journals, both with famous authors, both stressing the - A second contribution came from the progress in tech- unacceptable in their authors view - nonlocal prediction nology. Optical fibers had been developed and installed of quantum physics [1, 6]. A lot has been written on the all over the world. And Mandel’s group at the Univer- EPR ”paradox” and I won’t add to this. I believe that sity of Rochester (where I held a one-year post-doc posi- Einstein’s reaction is easy to understand. Here is the man tion and first met with optics) applied parametric down- who turned physics local, centuries after Newton wrote conversion to produce entangled photon pairs [10]. This his alarming text, he is proud of his achievement and cer- was enough (up to the detectors) to demonstrate quan- tainly deserves to be. Now, only a few years latter, non- tum nonlocality outside the curiosity laboratory. In 1997 locality reappears! Today one should add that quantum my group at Geneva University demonstrated the vio- nonlocality is quite a different concept from Newtonian lation of Bell inequalities between two villages around nonlocality, but Einstein did not fully realize this. Geneva, see Fig. 1, separated by a little more than 10 What Einstein and his colleagues saw is that quan- km and linked by a 15km long standard telecom fiber tum physics describes spatially separated particles as one [11, 12] (since then, we have achieved 50km [13]). So global system in which the two particles are not logically quantum nonlocality became politically acceptable! But separated. What they did not fully realize is that this what is it? Let me introduce the concept using students does not allow for signaling, hence it is not in direct con- undergoing ”quantum exams”. flict with relativity. In the next section I’ll try to present this using modern terminology. Most physicists didn’t pay much attention to this aspect of quantum physics. A kind of consensus established 2 Another story happened to me while I was a young post-doc eager that this was to be left for future examination, once the to publish some work. In a paper [8] I wrote ”A quantum particle technology would be more advanced. The general feeling may disappear from a location A and simultaneously reappear in was that quantum nonlocality was nothing but a labora- B, without any flow in-between”. The referee accepted the paper under the condition that this outrageous sentence is removed. tory curiosity, not serious physics. This referee considered his paternalist attitude so constructive Young physicists may have a hard time to believe that that he declared himself to me: ”look how helpful I am to you” such an important concept, like quantum nonlocality, (admittedly, he was politically correct). 3

VI. QUANTUM EXAMS: ENTANGLEMENT suffices that they prepare for the exam well enough. Note that the quantum exam #2 under consideration here is Assume that two students, Alice and Bob, have to pass even easier than standard exams, as there is no notion of some exams. As always for exams, the situation is ar- correct or incorrect answers. All that is required is that ranged in such a way that the students can’t communi- Alice and Bob give consistent answers: it suffices that cate during the exam. Clearly however, they are allowed, they jointly decide in advance which answer to give for and even encouraged, to communicate beforehand. Alice each of the possible questions. and Bob know in advance the list of possible questions, Now, a central problem: Could Alice and Bob suc- they also know that this is a kind of exam allowing only ceed with certainty for such an exam #2 by other means, a very limited number of possible answers, often only a that is without jointly deciding the answers in advance? binary choice between yes and no. During the exam Al- Think about it. If you found an alternative trick, then, if ice receives one question out of the list, let’s denote it by you are a student, you should use your trick to pass the x; Bob receives question y. Finally, denote a and b Alice next examination: just apply your trick together with the 4 and Bob’s answers, respectively. Hence, an exam is a re- best student, you’ll get the same mark as him/her . And alization of a random process described by a conditional if you are a professor and found a trick, then you should probability function, often merely called a correlation: stop testing your student with standard exams! Well, of course, there is no other trick, at least none applicable to P (a,b x, y) (1) classical students. | Correlations that satisfy P (a = b x = y) = 1 are nec- Clearly, the choice of questions x and y are under the essarily of the form | professor’s control. However, as all professors know, the students’ answers a and b are not! This is similar to ex- P (a,b x, y)= X ρ(λ)Q(a x, λ)Q(b y, λ) (2) | | | periments: the choice as to which experiment to perform λ is under the physicists control, but not the answer given by Nature. for some probability function Q and some distribution ρ In the following, we shall consider three kinds of exams, of common strategy λ. Historically, the λ were called in order to understand what kind of constraints they set ”local hidden variables”, computer scientist call them on the correlation P (a,b x, y). ”shared randomness”; here the λ denote common strate- | gies. P (a = b x = y) = 1 is but one example of a local A. Quantum exam #1 correlation,| among infinitely many others. Relation (2) characterizes all local correlations. In this first kind of quantum exam Alice is asked to In summary, some exams require common strategies; tell which question is given to Bob, and vice-versa. This in other words, some observed correlations can’t be ex- is clearly an unfair exam! Why? Because Alice and Bob plained except by common causes. are not supposed to communicate. How could they then succeed with a probability greater than mere chance3? This simple example shows that prohibiting signaling al- C. Quantum exam #3 ready limits the set of possible correlation P (a,b x, y). | For example P (a,b x, y)= δ(a = y)δ(b = x) is excluded. The third kind of quantum exam is the most tricky Notice that a correlation| P (a,b x, y) is non-signaling if | and interesting. For (apparent) simplicity let’s restrict and only if its marginal probabilities are independent of the set of questions and answers to binary sets and let the other side input: Pb P (a,b x, y) is independent of y us label them by bits, ”0” and ”1”. In this exam Alice and P (a,b x, y) is independent| of x. Pa | and Bob are required to always output the same answer, except when they both receive the question labelled ”1” in which case they should output different answers. Note B. Quantum exam #2 that formally this exam requires that Alice and Bob’s data satisfy the following equality, modulo 2: a+b = x y. · The second kind of quantum exam is closer to standard This time it is not immediately obvious whether they can exams. Alice and Bob are simply requested to provide prepare a strategy that guaranties success. the same answer whenever they receive the same ques- Assume first that the strategy forces Alice to output tion. This is clearly feasible: we all expect that good an answer that depends only on her input x, i.e. Alice’s students give the same answer to the same question. It strategy is deterministic. But in such a case, whenever

3 Somewhat surprisingly there is a strategy such that the proba- 4 Admittedly, the danger is that both student that get the bad bility that both players succeed is 50%. mark! But, on average, the poor student improves. 4

Bob receives the question 1, he can’t decide on his output since it should depend on Alice’s question. Next, if Alice’s output is random, this is clearly of no help to Bob. Consequently there is no way for Alice and Bob to succeed with certainty. Let us emphasize that successfully completing this exam does not necessarily imply communication between Alice and Bob. Indeed, assume that somehow Alice and Bob’s data would always satisfy a + b = x y. Would this allow Alice to communicate to Bob, or· vice-versa? Well, it depends! If Alice’s output a is known to Bob, for instance they decide on a = 0 always, then whenever Bob receives y = 1, he can deduce Alice’s question from a + b = x y and from his output: x = b in the example. But if Alice’s· outcome is unknown to Bob, for instance if Alice outcome is merely a random bit, then the relation FIG. 1: Bernex and Bellevue are the two villages north and a + b = x y is of no help to Bob. We shall come back · south of Geneva between which our long-distance test of Bell to this concept of a non-signaling correlation satisfying inequality outside the lab was performed in 1997, section V. a + b = x y in section X C. The inset represent two player that toss coins, as explained in Let us define· the mark M of this quantum exam #3 section VII. In the real experiment the coins were replaced by as the sum of the success probabilities [14]: photons, the players by interferometers, their right and left hands by phase modulators and head/tail by two detectors. M = P (a + b = xy x =0,y = 0) The experimental results are similar to that of the game, with | weaker but still nonlocal correlations. + P (a + b = xy x =0,y = 1) | + P (a + b = xy x =1,y = 0) | + P (a + b = xy x =1,y = 1) (3) VII. COIN TOSSING AT A DISTANCE | It is not difficult to realize that the optimal strategy for Another way to present nonlocality to non-physicist Alice and Bob consists in deciding in advance on a com- friends is the following. Imagine two hypothetical players mon answer, independent of the questions they receive. that toss coins. The players are separated in space and With such a strategy they are able to achieve the mark toss their coin once per minute. They use their free-will M = 3. This is indeed the optimal mark achievable by to decide, for each toss, whether to use their right hand common strategies: or their left hand, independently of each other. And they mark all results (time, hand and head/tail) in a big black M 3 (4) ≤ laboratory notebook, see Fig. 1. This is an example of a Bell inequality: a constraint on After thousands of tosses, they get bored. Especially, correlations arising from common strategies. Interest- given that nothing interesting happens: for each of the ing Bell inequalities are those that can be violated by two players, heads and tails occur with a frequency of quantum physics. In the case of (4), if Alice and Bob 50%, independently of which hand they use. Hence, the share singlets, then they can obtain the mark MQP = players decide to go for a beer. There, in the bar, they 2+ √2 3.41. Tsirelson proved that this is the highest compare their notes and get very excited. Indeed, quickly mark achievable≈ using quantum correlations [15]. they notice that whenever at least one of them happened Accordingly, quantum theory predicts that some tasks to have chosen his right hand for tossing his coin, both can be achieved that can’t be predicted by any local me- players always obtained the same result: either both head chanical model, i.e. some exams are passed with higher or both tails. But whenever, by mere chance, they both marks than classically possible. The fact that such tasks chose the left hand, then they always obtained opposite were invented for the purpose of showing the superiority results: head/tail or tail/head. A very remarkable corre- of quantum physics doesn’t affect the conclusion. Still, lation! it is only once some useful and natural tasks were found, The observation of correlations and the development concretizing the superior power of quantum physics over all possible local strategies, that quantum nonlocality became accepted by the physics community5. view Letters. I bet that a phase transition happen in the early 1990’s, after Ekert’s paper on quantum cryptography. In 1997 I started a PRL with the sentence: ”Quantum theory is nonlocal.” and got considerable reactions to what was felt as a provocative 5 I wish someone establishes the statistics of the occurrences of statement; today the same statement can be found in many pa- the words ”Bell inequality” and ”nonlocality” in Physical Re- pers, not provoking any reaction. 5 of theoretical models explaining them is the essence of [25])6. All these results proclaim loudly: God plays the scientific method. This is true not only in physics, dice. Note how ironic the situation is: the conclusion but also in all other sciences, like geology and medicine ”God plays dice” is imposed on us by the experimental for instance. John Bell used to say ”correlations cry out evidence supporting quantum nonlocality and by Ein- for explanations!” [16]. stein’s postulate that no information can travel faster So, why are our two players that excited by the correla- than light. Indeed, as mentioned in sub-section VI C, a tion they observe? Note that locally, nothing interesting violation of (4) with deterministic outputs leads to sig- happens; in particular there is no way for one player to naling. Consequently, the experimental violation of (4) infer from his data which hand the other player chose. and the no-signaling principle imply randomness [26, 27]. Even if one player decides to always use the same hand, Actually, the situation is even more interesting: Not this has no effect on the statistics observed by his col- only does God play dice, but he plays with nonlocal dice! league. Consequently, this game and the observed cor- The same randomness manifests itself at several relation do not imply any signaling. So, why do we feel locations, approximating a + b x y better than pos- ≈ · that this is impossible? Actually, frankly, I do not know! sible with any local classical physics model. Classical correlations are always explained by either of A very small minority of physicists still refuse to ac- two kinds of causes. The first kind is ”signaling”, one cept quantum nonlocality. They ask (sometimes with player somehow informs or influences the other player. anger) How can these two space-time locations, out there, This is clearly not the case here, since we assumed the know about what happens in each other without any sort players were widely separated in space (for the physicists of communication ?. I believe that this is an excellent we may add ”space-like separated”). The second kind of question! I have slept with it for years [28]. I summarize causes for classical correlations is a common cause. For my conclusion in the next section. example all football players simultaneously stop running, because the umpire whistled. This kind of cause is precisely equivalent to the assumption of a common strat- IX. ENTANGLEMENT AS A CAUSE OF egy, as formalized by (2) and excluded for the present CORRELATION correlation by Bell’s inequality (4). Consequently, the correlation observed by our two players is of a different Quantum physics predicts the existence of a totally nature. The big surprise is that anything beyond the two new kind of correlation that will never have any kind of classical causes for correlation exists! This is what Ein- mechanical explanation. And experiments confirm this: stein and many others had a hard time to believe. But, Nature is able to produce the same randomness at several today, if one accepts this as a matter of theoretical pre- locations, possibly space-like separated. The standard diction and experimental confirmation, then the next big explanation is ”entanglement”, but this is just a word, question is ”why can’t the correlation observed by our with a precise technical definition [29, 30]. Still words hypothetical players not be observed in the real world?”. are useful to name objects and concepts. However, it re- Indeed, quantum physics (and tensor products of Hilbert mains to understand the concept. Entanglement is a new spaces) tell us that Bell’s inequality (4) can be violated, explanation for correlations. Quantum correlations sim- i.e. not all quantum correlations can be explained by one ply happen, as other things happen (fire burns, hitting a of the two kinds of classical causes for correlations, but wall hurts, etc). Entanglement appears at the same con- quantum physics does not allow correlations as strong as ceptual level as local causes and effects. It is a primitive observed by our hypothetical players. Still, this game is concept, not reducible to local causes and effects. Entan- illustrative of quantum nonlocality, as we shall elaborate glement describes correlations without correlata [31] in a in section X holistic view [32]. In other worlds, a quantum correlation is not a correlation between 2 events, but a single event that manifests itself at 2 locations. VIII. EXPERIMENTS: GOD DOES PLAY DICE, Are you satisfied with my explanation of what entan- HE EVEN PLAYS WITH NONLOCAL DICE glement is? Well, I am not entirely! But what is clear is that entanglement exists. Moreover, entanglement is incredible robust! The last point might come as a sur- Physics is an experimental science and experiments prise, since it is still often claimed that entanglement is have again and again supported the nonlocal predic- as elusive as a dream: as soon as you try to talk about tions of quantum theory. All kind of experiments it, it evaporates! Historically this was part of the suspi- have been performed, in laboratories [18] and outside cion that entanglement was not really real, nothing more [11, 12, 19], with photons and with massive particles [20], with independent observers to close the locality loophole [11, 12, 17, 19, 21], with quasi-perfect detectors [20] to close the detection loophole, with high precision timing 6 The conclusion that follows from all these experiments is so im- to bound the speed of hypothetical hidden communica- portant for the physicist’s world-view, that an experiment clos- tion [22], with moving observers to test alternative mod- ing simultaneously both the locality and the detection loophole els [23] (multi-simultaneity [24] and Bohm’s pilot wave is greatly needed. 6 than some exotic particles that live for merely a tiny or as a NL-machine (a Non-Local machine7). The idea fraction of a second. But today we see a growing num- of these terminologies is to emphasize the similarities be- ber of remarkable experiments mastering entanglement. tween quantum measurements on 2 maximally entangled Entanglement over long distances [11, 12, 13, 19, 33], en- qubits and the correlation (5): in both cases the outcome tanglement between many photons [34] and many ions is available as soon as the corresponding input is given [35], entanglement of an ion and a photon [36, 37], en- (Alice knows a as soon as she inputs x into her part of the tanglement of mesoscopic systems (more precisely entan- machine and similarly Bob knows b as soon as he inputs glement between a few collective modes carried by many y, there is no need to wait for the other’s input) and in particles) [38, 39, 40], entanglement swapping [41, 42, 43], both the quantum and the PR-box cases the ”machine” the transfer of entanglement between different carriers can’t be used more than once (once Alice has input x, [44], etc. she can’t change her mind and give another input). No- In summary: entanglement exists and is going to affect tice a third nice analogy, neither the quantum nor the future technology. It is a radically new concept, requiring NL machines allow for signaling. Indeed, in all cases the new words and a new conceptual category. marginals are pure noise, independently of any input. Note that quantum physics is unable to produce the PR correlation (5). Indeed, this correlation violates the X. FROM QUANTUM NONLOCALITY TO Bell inequality (4) up to its algebraic maximum, M = 4, MERE NONLOCALITY while Tsirelson’s theorem [15] states that quantum correlations are restricted to M 2+ √2. However, the correlation (5) is much simpler≤ than quantum correla- So far we have seen that quantum physics produces tions, while sharing many of their essential features. In nonlocal correlations. And so what? Ok, this can be used particular (5) is nonlocal but non-signaling. for Quantum Key Distribution and other Quantum Infor- mation processes, but that doesn’t help much to under- In order to get some deeper understanding of the power stand non-locality. Conceptually, one would like to study of this hypothetical machine (5) as a conceptual tool, let non-locality without all the quantum physics infrastruc- us consider 3 properties of quantum correlations (many ture: Hilbert spaces, observables and tensor products. further nice aspects can be found in [47, 48, 49]). First Not too surprisingly, once the existence of non-locality we shall consider the so-called quantum no-cloning theo- was accepted, the conceptual tools to study it came very rem and see that it is actually not a quantum theorem, naturally. Actually, the tools were already there, in the but a no-signaling theorem. The next natural step is mathematics [45] and even the physics [26, 27] litera- to analyze quantum cryptography, whose security is of- ture, waiting for a community to wake up! The basic ten said to be based on the no-cloning theorem, and as tool is simple, doesn’t require any knowledge of quan- we would expect by now, we shall find ”non-signaling tum physics and allows one, so to say, to study quan- cryptography”. Finally, we consider the question of the tum nonlocality ”from the outside”, i.e. from outside the communication cost to simulate maximal quantum cor- quantum physics infrastructure. relation. But before all this we need to recall some facts about non-signaling correlations. Let us go back to the quantum exam #3 (subsection VI C). Assume that Alice and Bob are not restricted by quantum physics, but only restricted by no-signaling. Consequently, they would fail the quantum exam #1. A. The set of non-signaling correlations But under this mild no-signaling condition they could perfectly succeed in the quantum exam #3: Alice and Let us consider the set of all possible bi-partite correla- Bob would each output a bit which locally looks perfectly tions P (a,b x, y), where the inputs are taken from finite random and independent from their inputs - hence there alphabets x| and y and similarly for the outputs a would be no signaling - yet their 2 bits would satisfy and b , and{ } which{ are} non-signaling: { } a + b = x y, exactly as in the coin tossing game of { } section VII.· The hypothetical ”machine” that produces X P (a,b x, y)= P (a x) is independent of y (6) precisely this correlation is a basic example of the kind | | b of conceptual tools we need to study nonlocality without quantum physics. Formally, the correlation function is defined by: X P (a,b x, y)= P (b y) is independent of x (7) | | 1 a P (a,b x, y)= δ(a + b = x y) (5) | 2 · where the δ(z1 = z2) function takes value 1 for z1 = z2 7 and value 0 otherwise. A ”machine” is a physicists’s terminology for an input-output black-box that is not necessarily mechanical. I believe that this The correlation (5) is often referred to as a PR-box, to terminology appeared in the quantum physics context with the recall the seminal work by Popescu and Rohrlich [26, 27], ”optimal cloning machines” introduced by Buzek and Hillery [46] 7

a+b=xy 1 elementary symmetry (flip an input and/or an output). Although these polytopes live in an 8-dimensional space8, their essential properties can be recalled from the simple geometry of figure 2. Ö2-1 QM 0 facet B. No-cloning theorem corresponding isotropic to 3 P £ 3 correlations Details can be found in [47], as here we would merely like to present the intuition. Let us assume that Alice (input and output bits x and a, respectively) shares the correlation (5) both with Bob (bits y and b) and with polytope of Charly (bits z and c): a + b = xy and a + c = xz. Note local correlations that this situation is different from the case where Alice would share one ”machine” with Bob and share another independent ”machine” with Charly: in the situation under investigation Alice holds a single input bit x and a FIG. 2: Geometrical view of the set of correlations. The single output bit a. We shall see that the assumption bottom part represents the convex set (polytope) of local that Alice’s input and output bits x and a are correlated correlations, with the upper facet corresponding to the Bell both to Bob and to Charly leads to signaling. Hence in inequality (4). The upper triangle corresponds to the non- a Universe without signaling, Alice can’t share the cor- local non-signaling correlations that violate the Bell inequal- relation (5) with more than one partner: the correlation ity. The smooth thin curve limits the correlations achievable can’t be cloned. by quantum physics. The top of the triangle corresponds to the unique non-signaling vertex above this Bell inequality, i.e. In order to understand this, assume that Bob and to the non-local PR machine (5). The thin vertical line Charly come together, input y = 1 and z = 0, and add represents the isotropic correlations (8) with the indication their output bits b + c. According to the assumed corre- of some of the values of pNL. lations and using the modulo 2 arithmetic a + a = 0, one gets: b + c = a + b + a + c = xy + xz = x. Hence, they could determine from their data that Alice’s input bit is A priori this set looks huge. But it has a nice struc- x, i.e. Alice could signal to them! ture. First, it is a convex set: convex combinations of A natural question is how noisy should the correlation non-signaling correlations are still non-signaling. Sec- (5) be to allow cloning? The answer is interesting: as long ond, there are only a finite number of extremal points as the Alice-Bob correlation violates the Bell inequality (mathematician call such sets polytopes and the extremal (4), the Alice-Charly correlation can’t violate it; if not point vertices); accordingly every non-signaling correla- there is signaling. tion can be decomposed into a (not necessarily unique) We have just seen that the CHSH-Bell inequality (4) convex combination of extremal points. This is analog is monogamous, like well kept secrets. Let’s now see that to quantum mixed states that can be decomposed into this is not a coincidence! convex mixtures of pure states. Among the non-signaling correlations are the local ones, i.e. those of the form (2), analog to separable C. Non-signaling cryptography quantum states. The set of local correlations also forms a polytope, a sub-polytope of the non-signaling one. More- In 1991 Artur Ekert’s discovery of quantum cryptogra- over all vertices of the local polytope are also vertices of phy [9] based on the violation of Bell’s inequality changed the non-signaling polytope, see Fig. 2 [48]. The facets of the (physicist’s) world: entanglement and quantum non- the local polytope are in one-to-one correspondence with locality became respectable. Now, as we shall see in this all tight Bell inequality. subsection, the essence of the security of quantum cryp- Let us illustrate this for the simple binary case (which tography does not come from the Hilbert space structure is in any case the only one we need in this article), i.e. of quantum physics (i.e. not from entanglement), but is a,b,x,y are 4 bits. In this case, it is known that there due to no-signaling nonlocal correlation! The fact that are only 8 non-trivial Bell inequalities (i.e. not counting quantum physics offers a way to realize such correlation the trivial inequalities of the form P (a,b x, y) 0), i.e. makes the idea practical. However, if one would find any only 8 relevant facets of the local polytope.| Interestingly,≥ Barret and co-workers [48] demonstrated that the non- signaling polytope has only 8 vertices more than the local polytope, exactly one per Bell inequality! Each of these 8 8 More precisely, 8 is the dimension of the space of non-signaling vertices is equivalent to the PR correlation (5), up to an correlations [50]. 8 other way to establish such no-signaling nonlocal corre- correlated: lations (a way totally unknown today), then this would 1+ pNL 1 1 pNL 1 equally well serve as a mean to establish cryptographic P (a,b x, y)= δ(a+b = x y)+ − (8) keys [51]. | 2 2 · 2 4 Let us emphasize that the goal is to assume no restric- For pNL > 0 this correlation violates the inequality (4), tion on the adversary’s power, i.e. on Eve, except no- for pNL √2 1, it can be realized by quantum physics. 9 signaling [52]. Obviously, if one assumes additional re- Can Alice≤ and− Bob exploit such a correlation for crypto- strictions on Eve, like restricting her to quantum physics, graphic usage secure against an arbitrary adversary who then Alice and Bob can distill more secret bits from their is not restricted by quantum physics, but only restricted data. But qualitatively, the situation would remain un- by the no-signaling physics? The full answer to this fas- changed. cinating question is still unknown. However, there is an Assume that two partners, Alice and Bob, hold devices optimistic answer if one assumes that Eve attacks each that allow them to each input a bit (make a binary choice realization independently of the others, the so-called in- of what to do, e.g. which experiment to perform) and dividual attacks. In such a case, one may assume that each receives an output bit (e.g. a measurement result). Eve does actually distribute the apparatuses to Alice and This can be cast into the form of an arbitrary correla- Bob. Some apparatuses are ordinary local ones, for these tion: P (a,b x, y), with a,b,x,y four bits. Assume fur- Eve knows exactly the relation between the input and | thermore that the devices held by Alice and Bob do not output bits, both for Alice and for Bob. For example, allow signaling. This simple and very natural assump- Eve sends to Alice an apparatus that always outputs a tion suffices to give a nice structure to the set of corre- 0, and to Bob an apparatus that outputs the input bit: lations P (a,b x, y): as we recall in subsection X A this b = y. In this example Eve knows Alice’s bit a, but | set is convex and has a finite number of extreme points, doesn’t know Bob’s bit. For some local pairs of appara- called vertices. The nice property is that any correlation tuses Eve knows both a and b, or b but not a. But, if P (a,b x, y) can be decomposed into a convex combina- the Alice-Bob correlation (8) violates the Bell inequality | tion of vertices, hence once one knows the vertices one (4), i.e. if pNL > 0, then Eve must sometimes send to knows essentially everything. If the correlation is local, Alice and Bob the apparatuses that produce the max- i.e. of the form (2), then it is not useful for cryptogra- imal nonlocal correlation a + b = xy 10, in which case phy; indeed the adversary Eve may know the strategy she knows nothing about Alice and Bob’s output bits a λ. Hence, let’s assume that P (a,b x, y) violates the Bell and b. A detailed analysis can be found in [51]. Here | inequality (4). Consequently P (a,b x, y) lies in a well we merely recall the result. For pNL > 0.318 the Shan- defined corner of the general polytope,| a sub-polytope. non mutual information between Alice and Bob is larger Barrett and co-workers found that this sub-polytope has than the Eve-Bob mutual information [51]. Hence for only 9 vertices [48], 8 local ones for which M = 3 and pNL > 0.318 Alice and Bob can distil a cryptographic only one nonlocal vertex, that corresponding to our con- secret key out of their data, secure even against an hypo- ceptual tool, i.e. to a + b = xy, for which M = 4, see thetical post-quantum adversary, provided this adversary Fig. 2. is still subject to no-signaling. In the case that Alice and Bob are maximally corre- Actually, in [51] we worked out a 2-way protocol for lated (maximally but non-signaling!), i.e. their correla- key distillation valid down to pNL > 0.09. There, it is tion correspond to the nonlocal vertex of Fig. 2, it is also proven that the intrinsic information is positive for intuitively clear that the adversary Eve can’t be corre- all positive pNL. It is thus tempting to conjecture that lated neither to Alice, nor to Bob, by the no-cloning ar- secret key distillation is possible if and only if the Bell gument sketched in the previous subsection. Hence, in inequality is violated11. such a case Alice and Bob receive from their apparatuses perfectly secret bits. However, these bits are not always correlated: when x = y = 1 they are anti-correlated. But this can be easily fixed by the following protocol. After 10 One may think that Eve should sometimes send a weakly non- Alice and Bob received their output bits, Alice announces local machine. But all such correlations are convex combinations publicly her input bit x and Bob changes his output bit to of local and fully non-local NL-machines. Hence, it is equivalent for Eve to always send either a local or a NL-machine, with b′ = b + xy. Now Alice and Bob are perfectly correlated ′ appropriate probabilities. and Eve still knows nothing about a and b . 11 In [47] we proved that a correlation P (a, b|x,y) is nonlocal iff Consider now that Alice and Bob are non maximally any possible non-signaling extensions P (a, b, e|x,y,z) has positive Alice-Bob condition mutual information, conditioned on Eve, I(A, B|E), i.e. has positive intrinsic information. This nicely complements the similar result that holds for entangled quantum states and purifications [53]. In [51] we proved that 9 No-signaling should be understood here as in the previous sub- the same relation between nonlocality and positive intrinsic in- section on the no-cloning theorem. That is, even if two parties formation does also hold when Alice announces her input and joint, for example Eve and Bob come together, then they should Bob adapts his output in such a way as to maximize his mutual not be able to infer any information about the third party’s input, information with Alice. Proving this in full generality would be e.g. Eve and Bob should not have access no Alice’s input. a marvellous result. 9

Another beautiful result is the observation of an infor- D. Cost of simulating quantum correlations mation gain versus disturbance relationship, very similar to that of quantum physics, based on Heisenberg’s un- Among the many contributions of computer science to certainty relations [54]. Let us analyze separately the quantum information is the beautifully simple question cases where Alice announces x = 0 and x = 1, and de- (actually anticipated by Maudlin [58]): what is the cost of note the respective Alice-Bob error rates QBERx and the simulating quantum correlations? More precisely, Gilles Eve-Bob mutual informations Ix(B, E), i.e. QBERx = Brassard, Richard Cleve and their student Alain Tapp P (a = b) x, y) and Ix(B, E)= H(B x) H(B E, x). Py 6 | | − | [59], and independently Michael Steiner [60], asked the Remarkably, I0(B, E) is a function of only QBER1 and question: How many bits must Alice and Bob exchange 12 I1(B, E) of QBER0 : information gain for one input in order to simulate (projective) measurement outcomes necessarily produces errors for the other input, in anal- performed on quantum systems? The question concerns ogy with the quantum case where information gain on the communication during the measurement simulation, on basis necessarily perturbs information encoded in a clearly there must have been prior agreement on a com- conjugated basis! mon strategy. If the systems are in a separable state, no communication at all is needed. On the contrary, if the state allows measurements that violate a Bell inequality, i.e. if the state has quantum nonlocality, then it is impossible to simulate it without some communication or some other nonlocal resources. To conclude this subsection, let us emphasize that the distribution of the correlation (8) by quantum means re- For the simplest case of two 2-level systems (2 qubits), quires a protocol that differs from the famous BB84 pro- this game assumes that Alice and Bob receive as input ~ tocol [55]. Indeed, the data obtained by Alice and Bob any possible observable, i.e. any vector ~a and b of the following the BB84 protocol do not violate any Bell in- Poincarésphere. And they should output one bit, corresponding to the binary measurement outcome ”up” or equality, hence the BB84 protocol is not secure against 1 a non-signaling post-quantum adversary. Indeed, even ”down” in the physicist’s spin 2 language. A simple way the noise-free BB84 data can be obtained from quantum to simulate the quantum measurements is that Alice com- measurements on a separable state in higher dimension. municates her input ~a to Bob and outputs a predeter- The additional dimension could, for the example of po- mined bit (predetermined by Alice and Bob’s common larization coding, be side-channels due to accidental ad- strategy). But communicating a vector corresponds to ditional wavelength coding. Consequently, standard se- infinitely many bits! My first intuition was that there curity proofs [56, 57] must make assumptions about the is no way to do any better, after all the input space is a dimension of the relevant Hilbert spaces (accordingly, no continuum, quite the contrary to the case of Bell inequal- security proof of quantum key distribution is uncondi- ities where the input space is finite, usually even limited tional, contrary to widespread claims). But it is easy to to a binary choice. Yet, Brassard and co-workers came adapt the BB84 protocol, it suffices that Alice measures out with a model using only 8 bits of communication! the physical quantities corresponding to the Pauli ma- What a surprise: is entanglement that cheap? But this was only a start. Steiner published a model valid only for trices σz or σx, depending on her input bit value 0 or 1, respectively, exactly as in BB84, and Bob measures vectors lying on the equator of the sphere, but this model in the diagonal bases: σ+45o and σ−45o for y = 0 and was easy to generalize to the entire sphere [61]: it uses y = 1, respectively. In this way Alice and Bob’s data only 2 bits! 2 bits, like in dense coding and teleporta- are never perfectly correlated, but they can violate the tion: that should be the end, I thought! But, yet again, Bell inequality and be thus exploited to distil a secret key I was wrong. Bacon and Toner produced a model using valid even against post-quantum adversaries. Note that one single bit of communication [62]. Well, by now we the violation of a Bell inequality guarantees that no side should be at the limit, isn’t it? But actually, not quite! channels accidentally leak out information. Furthermore, Let’s come back to the real central question: How does in this protocol, that we like to call the CHSH-protocol, Nature manage to produce random data at space-like in honor of the 4 inventors [14] of the most useful Bell separated locations that can’t be explained by common inequality (actually equivalent to (4)), Alice announces causes? The idea that Nature might be exploiting some her input bit x, i.e. her basis as in BB84, but Bob doesn’t hidden communication (hidden to us, humans) is inter- speak, he always accepts and merely flips his bit in case esting. With my group at Geneva University we spent x = y = 1. In summary, in the CHSH protocol Alice and quite some time trying to explore this idea, both exper- Bob use all the raw bits, however their data are initially imentally and theoretically. We could set experimental noisier than in the BB84 protocol. bounds of the speed of this hypothetical hidden communication [22]. We also investigated the idea that each observer sends out hidden information about his result at arbitrary large speeds as defined in its own inertial 12 Precisely one has: I0(B,E) = 2 · QBER1 and I1(B,E) = 2 · reference frame [23]. The measured bounds on the speed QBER0. of the hypothetical hidden communication were very high 10

and the latter assumption contradicted by experiments. lation model for maximally entangled qubits using the Also our theoretical investigation cast serious doubts on PR-box. It would be astonishing if partially entangled the existence of hidden communication. Analyzing sce- state could not be simulated in a time-symmetric way narios involving 3 parties we could prove that if all quan- [69]. tum correlations would be due to hidden communication, then one should be able to signal (i.e. the hidden communication do not remain hidden) [63, 64]! Hence, the only XI. CONCLUSION remaining alternative is that Nature exploits both hidden communication and hidden variables: each one sep- The history of non-locality in physics is fascinating. arately contradicts quantum theory, but both together It goes back to Newton (section II. It first acceler- could explain quantum physics. However, this seems ated around 1935 with Einstein’s EPR and Schrödinger quite an artificial construction. Hence, let’s face the sit- cat’s papers. Next, it slowly evolved, with the works of uation: Nature is able to produce nonlocal data without John Bell, John Clauser and Alain Aspect among many any sort of communication. But is she doing so using others, from a mere philosophical debate to an experi- all the quantum physics artillery? Aren’t there logical mental physics question, or even to experimental meta- building blocks of nonlocality? A partial answer follows. physics as Abner Shimony nicely put it [70]. Now, during Let us come back to the problem of simulating quan- the last decade, it has run at full speed. Conceptually tum measurements, but instead of a few bits of commu- the two major breakthroughs were, first Artur Ekert’s nication let us give Alice and Bob a weaker resource: one 1991 PRL which strongly suggests a deep link between instance of the nonlocal machine a + b = xy. That this non-locality and cryptography, section XC. The second is indeed a weaker resource follows from the observation breakthrough, in my opinion, is the PR-box, section X A, that the correlation a + b = xy can’t be used to commu- the understanding that non-signaling correlations can be nicate any bit, but that by sending a single bit one can analyzed for themselves, without the need of the usual easily simulate the nonlocal correlation (since Alice’s in- Hilbert space artillery, thus providing a simple concep- put is only a bit x, it suffices that she communicates it to tual tool for the unravelling of quantum non-locality. We Bob). The nice surprise is that this elementary resource have reviewed that the no-cloning theorem, the uncer- is sufficient to simulate any pair of projective measure- tainty relation, the monogamy of extreme correlation and ments on any maximally entangled state of two qubits! the security of key distribution, all properties usually as- For the proof the reader is referred to the original arti- sociated to quantum physics are actually properties of cle [65] and to the beautiful account in [66] where the any theory without signaling, section X. In particular relations between all these models are presented. we emphasized that the second breakthrough, the PR- box, allows one to confirm the first breakthrough: there The above results are very encouraging. One can gets is an intimate connection between violation of a Bell in- the feeling that, at last, one can start understanding non- equality and security of quantum cryptography. locality without the Hilbert space machinery, that, at And relativity, can it be considered complete? Well, last, one can study quantum physics from the outside, if nonlocality is really real, as widely supported by the i.e. from the perspective of future physical theories (as- accounts summaries in this article, then all complete the- suming these will keep Einstein’s no-signaling constraint) ories should have a place for it. Hence, the question is: and no longer from the perspective of the old classical me- ”Does relativity hold a place for non-signaling nonlocal chanical physics. But there is still a lot to be done! For correlations?”. instance, it is surprising (and annoying in my opinion) that one is still unable to simulate measurement on partially entangled states using the nonlocal correlation (ac- Acknowledgment tually we could prove that this is impossible with a single instance of the correlation, but there is hope that one can This article has been inspired by talks I gave in 2005 at the simulate partially entangled qubit pairs with 2 instances IOP conference on Einstein in Warwick, the QUPON confer- [67]). Let me emphasize that all of today’s known simu- ence in Vienna, the Annus Mirabilis Symposium in Zurich, le lation models for partially entangled qubits include some séminaire de l’Observatoire de Paris and the Ehrenfest Collo- 13 sort of communication [62], let’s say from Alice to Bob. quium in Leiden. This work has been supported by the EC Consequently, in all these models Bob can’t output his under projects RESQ and QAP (contract n. IST-2001-37559 results before Alice was given her input. This contrasts and IST-015848) and by the Swiss NCCR Quantum Photon- with the situation in quantum measurements where Bob ics. doesn’t need to wait for Alice (he does not even need to know about the existence of Alice) and with the simu-

13 Using the reduction of an OT-box (Oblivious Transfer to a PR- box) [68] one can simulate any 2-qubit state with one OT-box. 11

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