# GAP Optique Geneva University 1 Quantum Nonlocality: How does Nature do it? And what can we do with it? Nicolas Gisin GAP, University of Geneva, Switzerland.

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GAP Optique Geneva University 1 Quantum Nonlocality: How does Nature do it? And what can we do with it? Nicolas Gisin GAP, University of Geneva, Switzerland 1.The scene: what is nonlocality ? 2.How does Nature perform the trick ? 3.What can nonlocal correlations do for us ? 4.Deterministic nonlocal variables 5.Hidden influences

GAP Optique Geneva University 2 AliceBob   x y a b P  (a,b|x,y) ≠ P  (a|x)·P  (b|y) The events at Alice and Bob’s sides are not independent ! It seems that somehow the two sides are coordinated or “interact” !?! (but without signaling) Spatially separated systems are not logically separated.  Quantum Physics is nonlocal

GAP Optique Geneva University 3 During my early carrier as a physicist... when I was about 6 months old… I learned the hard way that in order to interact with an object I had either to crawl to it or to throw something at it.

GAP Optique Geneva University 4 Conclusion: Bell inequalities AliceBob x y a b 1.locality: P (a,b|x,y) = P (a|x)·P (b|y) where =physical state of the systems according to any possible future theories. 2.a Alice can freely choose her input x and read the outcome a, similarly for Bob. 2.b x & y are independent of I(x: )=I(y: )=0 Assumptions: N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255, Found. Phys. 42, 80 (2012) NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009

GAP Optique Geneva University 5 AliceBob x y a b T oday, we don’t know : P(a,b|x,y) =  d P (a|x)·P (b|y)  Bell inequality Bell inequalities 1. Violation:  all future theories have to be nonlocal. 2. No violation:  there is a separable quantum state and measurements reproducing P(a,b|x,y). PRL 97, 120405, 2006; PRL 100, 210503, 2008. N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255, Found. Phys. 42, 80 (2012) NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009

GAP Optique Geneva University 6 AliceBob x y a b Don’t think of as an old fashion local hidden variable. Think of as the physical state of the systems as described by any possible future theory. Could be the state of the entire universe, except that can’t determine x nor y. Studying Bell’s inequality tells us something about any possible future theory compatible with today’s experimental observations. N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255, Found. Phys. 42, 80 (2012) NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009

GAP Optique Geneva University 7 AliceBob x y a b No signalling Bob’s statistics are independent of Alice’s input x. Alice’s statistics are independent of Bob’s input x. N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255, Found. Phys. 42, 80 (2012) NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009

GAP Optique Geneva University 8 AliceBob x y a b The only assumption in the derivation of Bell inequality, besides the locality assumption, is that x,y,a and b are classical variables. That is that one directly access them, copy, memories and broadcast them. Non realism seems to me not an alternative to nonlocality. And what could “local non realism” mean ?!? Non realism ? N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255, Found. Phys. 42, 80 (2012) NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009

GAP Optique Geneva University 9 The CHSH-Bell inequality x,y,a,b = bits E(x,y)=p(a=b|x,y)-p(a≠b|x,y) E(0,0)+E(0,1)+E(1,0)-E(1,1) Alice Bob X=0.. Y=0 x=1.. Y=1  S CHSH  E(0,0) + E(0,1) + E(1,0) - E(1,1)  2 But S CHSH (Quantum) = 2  2  

GAP Optique Geneva University 10 Nonlocality without Signaling  It is very difficult to think of Nonlocality without some sort of communication “behind the scene”.  Yet, quantum physics is nonlocal, but doesn’t allow for “signaling”:  The statistics on Bob’s side do not depend on Alice’s input: p(b|x,y)  a p( a,b|x,y) = p(b|y)  And similarly: p( a |x,y) = p( a |x).

GAP Optique Geneva University 11 No-cloning and no-signaling  Perfect cloning would allow one to exploit nonlocal correlations for signaling: * Alice Bob } clones M Source of entangled particules  Interestingly, optimal-but-imperfect quantum cloning is precisely at the no-signaling limit ! NG, Phys. Lett. A 242, 1 (1998) * Alice Bob } Noisy clones M Source of entangled particules Laser amplifier: stimulated & spontaneous emissions signalingNO signaling

GAP Optique Geneva University 12 Nature is Nonlocal (but without signaling). And so?  Physics: Amazing ! How can these two locations out there in space-time know about each other ?  According to Quantum Physics Quantum correlation just happen, somehow from outside space-time : there is no story in space-time that can tell us how it happens.  Computer Science: What can one do with nonlocal non-signaling (quantum) correlations ?

GAP Optique Geneva University 13 a + b= x.y AliceBob x  {0,1} a  {0,1} b  {0,1} y  {0,1} PR-box: E(0,0) + E(0,1) + E(1,0) - E(1,1) = 4 Prob(a=1|x,y) = ½, independent of y  no signaling A single bit of communication suffice to simulate the PR-box (assuming shared randomness). But the PR-box does not allow any communication. Hence, the PR-box is a strictly weaker resource than communication. Found.Phys. 24, 379, 1994

GAP Optique Geneva University 14  The PR-box is the unique extremal nonlocal box for binary inputs and outputs.  Why is Quantum Physics not more nonlocal than it is ?!?  What could one do, had we PR-boxes ?

GAP Optique Geneva University 15 No-cloning from no-signaling x a y b z c a+b=x.y a+c=x.z a+b=x.y b y  b+c = x.(y+z)  If y+z=1, Bob and Charly can deduce Alice’s input out of their outputs  Signaling !!!  In a non-signaling world, cloning of PR-boxes is impossible A noisy PR box can be cloned iff it is noisy enough to be local.

GAP Optique Geneva University 16 From non-signaling nonlocality to cryptographic keys  The quantum no-cloning theorem is the basis for Quantum Key Distribution.  Since no-cloning holds for all non-signaling nonlocal correlations, QKD should be secure against any non-signaling adversary, even post- quantum (i.e. mastering technologies described by any future theory, provided it is non- signaling).

GAP Optique Geneva University 17 Basics of QKD Alice Bob 0 1 x=0 or 1 a y=0 or 1 b After publicly announcing a fair sample of their data, Alice and Bob’s information is entirely contained in the conditional probability p(a,b|x,y) If E(0,0)+E(0,1)+E(1,0)-E(1,1) is large enough, then Alice and Bob can distil a secret key secure against any possible individual attack by any non-signaling adversary.

GAP Optique Geneva University 18 The no-signaling polytope polytope of local correlations a+b=xy QM J.Barrett et al, PRL’05 PR-box Only one new vertex per CHSH facet of the local polytope facet corresponding to the CHSH-Bell  : E(0,0)+E(0,1)+ E(1,0)-E(1,1)  2 vertex: local deterministic correlation

GAP Optique Geneva University 19 A bit more on the Theory side Studying nonlocal correlations from outside the Hilbert space P(a,b|x,y) can be considered as a vector and represented as a point in a vector space: Local Bell inequality Quantum No signalling New question: why are quantum correlations not more nonlocal ?

GAP Optique Geneva University 20

GAP Optique Geneva University 21 Deterministic nonlocal hidden variables e.g. ={ ,r a,r b }. Alice Bob  (-) t=0 t=1 t=2  =F AB (, )  =S AB (,, )  

GAP Optique Geneva University 22 Alice Bob  (-) t=0 t=1 t=2  =F AB (, )  =S AB (,, )   t’=2 t’=1 t’=0  =F BA (, )  =S BA (,, ) Deterministic nonlocal hidden variables

GAP Optique Geneva University 23 Alice Bob  (-) t=0 t=1 t=2  =F AB (, )  =S AB (,, )   t’=2 t’=1 t’=0  =F BA (, )  =S BA (,, ) Could there be, F AB, S AB, F BA and S BA s.t. F AB (, ) = S BA (,, ) ? Quantum correlations can’t be described with local variables, nor can they be described with deterministic nonlocal variables. Theorem: NO ! Proof: S BA would be independent of  locality  Bell inequality. Impossibility of covariant deterministic nonlocal hidden-variable extensions of quantum theory NG, PRA 83, 020102, 2011

GAP Optique Geneva University 24 Finite-Speed Hidden-Influence Explanations of Quantum Correlations lead to Signalling  A principle of continuity  Quantum nonlocality based on finite-speed causal influences leads to superluminal signalling, arXiv:1110.3795  Nature is nonlocal in the sense of discontinuous Nicolas Gisin J.-D. Bancal, S. Pironio, A. Acin, Y.-C. Liang, V. Scarani Group of Applied Physics, University of Geneva

GAP Optique Geneva University 25 A stone moved on the moon would immediately affect the gravitational field on earth. Newton’s Nonlocality How can these two locations out there in Space-time know about each other ?

GAP Optique Geneva University 26 Let’s read Newton’s words: That Gravity should be innate, inherent and essential to Matter, so that one Body may act upon another at a Distance thro’ a Vacuum, without the mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity, that I believe no Man who has in philosophical Matters a competent Faculty of thinking, can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain Laws, but whether this Agent be material or immaterial, I have left to the Consideration of my Readers. Isaac Newton Papers & Letters on Natural Philosophy and related documents Edited by Bernard Cohen, assisted by Robert E. Schofield Harvard University Press, Cambridge, Massachusetts, 1958

GAP Optique Geneva University 27 Signalling = non-physical communication  To send information one has to encode it into a physical support and send this support to the receiver. Landauer: Information is physical.  Any other way of communicating would be signalling; it would be non-physical communication.  Moreover, signalling would allow faster-than-light communication. But no-signalling is even more fundamental than relativity.

GAP Optique Geneva University 28 Principle of Continuity  Everything (mass, energy and information) propagates gradually and continuously through space as time passes.  Nothing jumps instantaneously from here to there (no instantaneous teleportation).  Correlations can have only two types of explanations. Either common local cause or influences at finite speed t x

GAP Optique Geneva University 29 Principle of Continuity Explanations of correlations by local common causes Explanations of correlations by an event influencing another one VariablesInfluences (hidden) LocalFinite speed Bell’s theoremThis talk Contradiction with quantum predictions Falsified explanation

GAP Optique Geneva University 30 Satigny – Geneva – Jussy N S E W Experimental lower bound on the speed of hidden influence Nature 454, 861, 2008 Cocciaro et al., PLA 375, 379, 2011

GAP Optique Geneva University 31 And so ?  The influence may merely propagate faster,  or may not exist at all.  2-party experiments will never be able to exclude hidden influences, only set lower bounds on its speed.

GAP Optique Geneva University 32 -causality  Assume that a hidden influence propagates at speed v < .  v can be larger than c (defined in a universal privileged frame).  Whenever an event happens, the rest of the universe is informed at speed v.  Whenever the hidden influence arrives on time, future events are correlated as predicted by QM.  Whenever the hidden influence doesn’t arrive on time, events can only be Bell-local correlated (i.e. correlated by local variables).  -causal predictions may differ from quantum theory  Bohm We shall see that hidden influence at finite speed leads to signalling at the macroscopic level, i.e. hidden influences at finite speed can’t remain hidden

GAP Optique Geneva University 33 -causality Light cone v-cone t priv. x privileged  

GAP Optique Geneva University 34 -causality leads to signalling t priv. x privileged  Alice Bob Charlie Dave   A B C D  ABD  is quantum  ACD  is quantum  BC  is local even if conditioned on A and D

GAP Optique Geneva University 35 Q predictions for v-causality BC|AD local  p(a,b,c,d|x,y,z,w)=p(a,d|x,y,z,w)  p(b|y, x,a,w,d, z)  p(c|z, x,a,w,d, y) V-causality / / / / Q prediction: where  for the state and setting of our paper, p(abc|x,y,z,w) is signalling iff a=1 & x=1 & z=0 p(bcd|x,y,z,w) is signalling iff z=1 & w=0 arXiv:1110.3795

GAP Optique Geneva University 36 Principle of Continuity Explanation of correlations by local common causes and influences Variables and Influences (hidden) Local and Finite speed Rest of this talk

GAP Optique Geneva University 37 -causality leads to signalling x priv.  Alice Bob Charlie Dave  A B C D x a yzw bcd binary inputs 0/1 binary outcomes  1 Theorem: If p(a,b,c,d|x,y,z,w) is non-signalling and p(b,c|y,z, a,x,d,w) is local for all a,x,d,w, then J  7 Where J = -3  A 1  -  B 0  -  B 1  -  C 0  - 3  D 0  -  A 1 B 0  -  A 1 B 1  +  A 0 C 0  + 2  A 1 C 0  +  A 0 D 0  +  B 0 D 1  -  B 1 D 1  -  C 0 D 0  - 2  C 1 D 1  +  A 0 B 0 D 0  +  A 0 B 0 D 1  +  A 0 B 1 D 0  -  A 0 B 1 D 1  -  A 1 B 0 D 0  -  A 1 B 1 D 0  +  A 0 C 0 D 0  + 2  A 1 C 0 D 0  + 2  A 0 C 1 D 1 

GAP Optique Geneva University 38 t priv. x privileged  Alice Bob Charlie Dave   A B C D J = -3  A 0  -  B 0  -  B 1  -  C 0  - 3  D 0  -  A 1 B 0  -  A 1 B 1  +  A 0 C 0  + 2  A 1 C 0  +  A 0 D 0  +  B 0 D 1  -  B 1 D 1  -  C 0 D 0  - 2  C 1 D 1  +  A 0 B 0 D 0  +  A 0 B 0 D 1  +  A 0 B 1 D 0  -  A 0 B 1 D 1  -  A 1 B 0 D 0  -  A 1 B 1 D 0  +  A 0 C 0 D 0  + 2  A 1 C 0 D 0  + 2  A 0 C 1 D 1 

GAP Optique Geneva University 39 t priv. x privileged  Alice Bob Charlie Dave  A B C D J = -3  A 0  -  B 0  -  B 1  -  C 0  - 3  D 0  -  A 1 B 0  -  A 1 B 1  +  A 0 C 0  + 2  A 1 C 0  +  A 0 D 0  +  B 0 D 1  -  B 1 D 1  -  C 0 D 0  - 2  C 1 D 1  +  A 0 B 0 D 0  +  A 0 B 0 D 1  +  A 0 B 1 D 0  -  A 0 B 1 D 1  -  A 1 B 0 D 0  -  A 1 B 1 D 0  +  A 0 C 0 D 0  + 2  A 1 C 0 D 0  + 2  A 0 C 1 D 1  Any v-causal model predicts the same value for J as QM -causal predictions differ from Q theory, but since J doesn’t contain any term involving B and C, the -causal prediction for J is merely the Q value. Moreover, in an experiment B and C do not need to be measured in the same run.  No B-C timing issue ! 

GAP Optique Geneva University 40 -causality leads to signalling Theorem: If p(a,b,c,d|x,y,z,w) is non-signalling and p(b,c|y,z, a,x,d,w) is local for all a,x,d,w, then J  7 Fact: there are quantum states and measurements predicting J>7 Consequence: Since any v-causal model predicts that p(b,c|y,z, a,x,d,w) is local, p(a,b,c,d|x,y,z,w) must be signalling. arXiv:1110.3795

GAP Optique Geneva University 41 -causality leads to supraluminal communication

GAP Optique Geneva University 42 Principle of Continuity Explications of correlations by local common causes Explications of correlations by an event influencing another VariablesInfluences (hidden) LocalFinite speed Bell’s theoremSignalling Contradiction with quantum predictions Contradiction with no faster than light communication Falsified explanation Nature doesn’t satisfy the principle of continuity Nature is nonlocal This talk

GAP Optique Geneva University 43 Conclusion  There is no spooky action at a distance: there is not a first event that influences a second event.  Quantum correlation just happen, without any time-ordering, somehow from outside space- time !  There is no story in space-time that tells us how nonlocal Q correlations happen. Shall scientists give up the Grand Enterprise of explaining how Nature does it? Explaining = telling stories  We need to enlarge our story-tool-box to explain nonlocal correlations arXiv:1110.3795

GAP Optique Geneva University 44 New tools for our story-tool-box Instead of claiming with Einstein that God doesn’t play dice, let’s ask why he plays dice. The answer is that so Nature can be nonlocal without signalling (without non-physical communication).  Let’s tell stories in which random events can manifest themselves at several locations,  i.e. stories that include nonlocal randomness.

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