Nonlocality (CH. I & II)

Transcription

1 Nonlocality (CH. I & II) Nicolas Gisin GAP, University of Geneva, Switzerland 1

2 x P Ψ (a,b x,y) P Ψ (a x) P Ψ (b y) y Alice Ψ a The events at Alice and Bob s sides are not independent! It seems that somehow the two sides are coordinated or interact!?! (but without signalling) Ψ b Bob Spatially separated systems are not logically separated. Quantum Physics is nonlocal 2

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. 3

4 Telekinesis: doesn t work! Likewise, telepathy doesn t work: no information can pass from one location to another without a physical support carrying this information. 4

5 x P Ψ (a,b x,y) P Ψ (a x) P Ψ (b y) y Alice Ψ a The events at Alice and Bob s sides are not independent! It seems that somehow the two sides are coordinated or interact!?! (but without signalling) Ψ b Bob How can these spatially separated systems coordinate without some sort of telepathy? Could we use such correlation to communicate? 5

6 Newton s nonlocality haut bas 6

7 Tout en tombant, la lune avance la lune tombe à côté de la terre 7

8 Tout en tombant, la lune avance la lune tombe à côté de la terre 8

9 F = G md M 2 9

10 A good question How does the moon know in which direction to fall? Does the moon use a kind of stick to feel where Earth is? Or does she send us some kind of balls? 10

11 Newton s Nonlocality A stone moved on the moon would immediately affect the gravitational field on earth. How can these two locations out there in Space-time know about each other? 11

12 Newton s Nonlocality A stone moved on the moon would immediately affect the gravitational field on earth. Had someone tested this prediction, he would: 1. have falsified Newton s theory, and 2. have found that gravity propagates at the speed of light 12

13 Newton already ask this important question: 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,

14 Einstein, the greatest mechanical engineer Today, thanks to Einstein, gravitation is no longer considered as a kind of action at a distance. A moon-quake triggers a bunch of gravitons that propagate through space and «informs» Earth. The propagation is very fast, but at finite speed, the speed of light, i.e. about 1 second from the moon to our Earth. 14

15 Einstein, the greatest mechanical engineer In 1905, Einstein also gave a description of Brownian motion: the statistics of collisions between invisible atoms and molecules support the atomic hypothesis: Still in 1905, Einstein gave a mechanical explanation of the photo-electric effect: 15

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17 Because of the propagation delay of the gravitons, the moon does fall straight on Earth center, but a bit next to it. 17

18 Experiment With sufficient technology, one could have proven Newton s theory of universal gravitaion wrong. One could even have found the speed of what was then a hidden influence But still today, this experiment is not feasible Let s look for qravitons that would explain quantum nonlocality 18

19 Assume a real influence propagating faster than light but with finite speed x y Alice Ψ a hidden influence Ψ b Bob 19

20 Assume a real influence propagating faster than light but with finite speed x y Alice Ψ a hidden influence Ψ??? Bob 20

21 Tests of Bell s inequality: Aspect s 1982 experiment The curiosity lab 21

22 Our photon-pair source, 1997 Laser 655nm P F KNbO 3 L energy-time entanglement output 1 output 2 crystal laser 45 cm lens filter 40 cm λp = 655 nm; λs, i = 1310 nm diode laser small and handy 40 x 45 x 15 cm 3 I pump = 8 mw 5 khz coincidences PRL 81, 3563 (1998) PRA 59, 4150 (1999) 22

23 Closing the locality loophole PRL 81, 3563 (1998) PRA 59, 4150 (1999) Phys. Lett. A 264, 103 (1999) 1 st setting 1 st setting Alice 2 nd setting Source * 2 nd setting The real-world lab Bob 23

24 Satigny Geneva Jussy N W E S In which frame should the events be simultaneous? 24 Nature 454, 861, 2008

25 Let s test this hypothetical preferred reference frame A B Alice and Bob, east-west orientation, perfect synchronization with respect to earth perfect synchronization w.r.t any frame moving perpendicular to the A-B axis in 12 hours all hypothetical privileged frames are scanned. Ph. Eberhard, private communication 25

26 Time (min) Nature 454, 861, Coincidences/60s

27 1 Visibility Visibility :00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:00 Time 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:00 Time Nature 454, 861,

28 Finite precision The «Speed of Quantum Information» V QI is V QI r t ' B ' B r t ' A ' A After a Lorentz transformation, one finds V QI c ( 1 β )( 1 ρ ) 2 ( ρ + β ) 1+ ( )( ) β 1 ρ ( ρ + β ) 2 To get a good bound on V QI, one should ensure a good alignment, and should upper bound β (the component of β parallel to the AB direction). T = 360 seconds δ α 5.8 o Two cases can be considered. For each case, there exists a period of time T, during which β (t) is upper-bounded by: α S N χ β tan χ > if tanα ρ ρ β tan χ < tanα β ( t) β = v/c is the relative speed of the Earth frame in the privileged frame, ρ = ct AB /r AB defines the alignment of the 2 detections in the Earth frame ωt β sin χ cos α cos χ sin α 2 ωt β cos χ sinα sin χ cosα cos 2 ( t) ρ 28 Nature 454, 861, 2008

29 Bound on V QI /c Bound assuming the Earth s speed is 300 km/s χ ( ) Bound assuming χ = 90 o 29 Nature 454, 861, 2008

30 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. With only 2 parties, the hypothetical hidden influence could remain hidden. 30

31 Could a real influence propagating faster than light but with finite speed remain hidden for ever? Or would any real influence propagating faster than light but with finite speed necessarily allow one to communicate faster than light? 31

32 Hidden influences alone would allow Alice faster-than-light communication Ψ ABC = GHZ = Bob Charlie If Alice doesn t measure, then Bob & Charlie's outcomes are independent. Perfect Synchronization No time for hidden influences If Alice first measures σ z and Bob & Charlie later also measure σ z, then Bob & Charlie s outcomes are always the same. V. Scarani, W. Tittel, H. Zbinden and N. Gisin, Phys.Lett. A 276, 1-7 (2000). V. Scarani & NG, Brazilian J. of Physics 35, (2005), quant-ph/ L.C. Ryff, arxiv: Ph. Eberhard, in "Q theory and pictures of reality", Ed W. Schommers, Spinger- Verlag, Heidelberg, pp ,

33 Hidden influences would allow signalling, i.e. communication without transmission Alice Ψ ABC = GHZ = Bob Charlie Perfect Synchronization No time for hidden influences An immediate objection is that whenever the hidden influence doesn t arrive on time, the outcome could be determined by local hidden variables. This corresponds to an explanation combining local hidden variables and finite-speed influences. 33

34 How does Nature perform the trick? How can these two locations out there in space-time know about each other? How does an event A know that it is nonlocally correlated to another event B? Who keeps track of who is entangled with whom? How does nature do it?, Science, 326, 1357,

35 Alice x λ a Assumptions: 1. locality: P λ (a,b x,y) = P λ (a x) P λ (b y) where λ=physical state of the systems according to any possible future theories. λ λ 2.a Alice can freely choose her input x and read the outcome a, similarly for Bob. 2.b x & y are independent of λ I(x:λ)=I(y:λ)=0 Conclusion: Bell inequalities N. Gisin, Non-realism : deep thought or a soft option? quant-ph/ , Found. Phys. 42, (2012) y λ b Bob 35

36 x Quantum exams y Alice Bob Joint conditional probability a P ( a, b x, y Events at 2 separated locations. Not under the professor s control ) b Settings (experimental conditions). Under the professor s control 36

37 Binary inputs and outputs: CHSH Let a0 and a1 denote Alice s outcomes for inputs x=0 and x=1, respectively. Similarly for b0 and b1. Assume a0,a1,b0,b1 {-1,1} Obviously: a0 (b0+b1) + a1 (b0-b1) = ±2 2 Hence: E(0,0)+E(0,1)+E(1,0)-E(1,1) 2 where P(a=b x,y)-p(a b x,y) This is the CHSH-Bell inequality. 37

38 The Bell Game Heure x a 9h h h h h h h06... a=0 x=0 x=1 38

39 Alice The Bell Game Bob x=0 x=1 y=0 y=1 a=0 Goal of the game: 1. Whenever x=0 or y=0, a=b 2. Whenever x=1 and y=1, a b b=0 Note: only the correlation between the outcomes a and b is important, the individual values of a and of b are irrelevant. Only by comparing a and b can Alice and Bob know whether they achieved the goal or not. 39

40 Alice The Bell Game Bob x=0 x=1 y=0 y=1 a=0 Goal of the game: 1. Whenever x=0 or y=0, a=b 2. Whenever x=1 and y=1, a b b=0 Note: The goal of the game can be sumaries in an equation: a + b = x y (modulo 2) The mathematics of the Bell game is trivial. 40

41 Alice The Bell Game Bob x=0 x=1 y=0 y=1 a=0 Goal of the game: 1. Whenever x=0 or y=0, a=b 2. Whenever x=1 and y=1, a b b=0 Score: - Repeat the game very often choosing the x,y at random. - Come together and compare the results. - For all combinations of choices (x,y) compute the rate of success - Add the 4 rates. 0 S = <a+b=x y> 4 41

42 How to win the Bell Game Alice x=0 x=1 y=0 Bob y=1 a=0 S = <a+b=x y> b=0 1. Communication from Alice to Bob and/or from Bob to Alice. 2. Agree in advance on some strategy. An example of a simple strategy is to decide to always produce the outcome 0. In this way S=3 Another example is that Alice always produces a=0, while Bob b=y. (x,y)=(0,0) ok, (x,y)=(0,1) no!, (x,y)=(1,0) ok, (x,y)=(1,1) ok S =3 42

43 How to win the Bell Game Alice x=0 x=1 y=0 Bob y=1 a=0 S = <a+b=xy> b=0 Alice has only 4 possible strategies (2 inputs & 2 outcomes: 2 2 =4): 1: a = 0 The strategy may change from minute 2: a = 1 to minute, but at each minute Alice 3: a = x uses one strategy. 4: a =1-x Bob has also only 4 possible strategies. Hence, together they have 4 4=16 combinations of strategies. All combinations give S=1 or S=3. S 3 S 3 Bell inequality 43

44 E(x,y) = P(a=b x,y)-p(a b x,y) =2 P(a=b x,y)-1 = 1-2 P(a b x,y) Hence, E(0,0)+E(0,1)+E(1,0)-E(1,1) = = 2(P(a=b 00)+P(a=b 01)+P(a=b 10)+ P(a b 11)) 4 =2<a+b=x.y>

45 Alice x λ a Don t think of λ as an old fashion local hidden variable. Think of λ as the physical state of the systems as described by any possible future theory. λ λ could be the state of the entire universe, except that λ 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/ , Found. Phys. 42, (2012) NG, Quantum nonlocality : how does nature do it?, Science, 326, 1357, 2009 λ y λ b Bob 45

46 2 ways of thinking of λ 1. λ is the state of the Universe (except that the inputs x and y are independent of λ): this is the standard way of thinking, though usually implicitly only 2. λ is carried by the particule: probably Bell s original idea. This leads to n-locality ( locality). 3. The difference appears only for more than 2 parties are involved in the game. 46