Spooky Action at a Distance

1. Spooky Action at a Distance Bell’s Theorem and the Demise of Local Reality Natalia Parshina Peter Johnson Josh Robertson Denise Nagel James Hardwick Andy Styve Bell's Theorem

2. Introduction • Einstein’s Belief • Bell’s Gedankenexperiment • Simplified Experiment • Full Version • Table 1 and 2 • Theoretical prediction of K • Table 1’ and 2’ • The demise of local reality • Simulation Bell's Theorem

3. Einstein’s Belief • Local Reality • Principle of Separability: • The outcome of experiment X and Y will be independent when information from X cannot reach Y. • Objective Reality: • philosophical perspective on reality. • Objects have existence independent of being known. Bell's Theorem

4. Postulates of Quantum Mechanics • Quantum system can be modeled by a complex inner product space: v = Cn • Evolution of quantum stated are described by unitary operators. • Quantum measurements are “described” by a finite set of projections acting on the state space being measured. • The state of a composite, multi-particle, quantum system formed from X1, X2, …,Xn is the tensor product of the set. Bell's Theorem

5. Postulates of Quantum Mechanics • Quantum system can be modeled by a complex inner product space: v=Cn Bell's Theorem

6. Postulates of Quantum Mechanics • Evolution of quantum states are described by unitary operators. • Example: A-1=AT Bell's Theorem

7. Postulates of Quantum Mechanics • Quantum measurements are “described” by a finite set of projections acting on the state space being measured. • Suppose the state of a system is: prior to observation, then P(m) = Bell's Theorem

8. Postulates of Quantum Mechanics Continued.. If result m occurs, the new state of the system will be given by: Bell's Theorem

9. Postulates of Quantum Mechanics • The state of a composite (multi-particle) quantum system formed from: is Bell's Theorem

10. Bell’s Gedankenexperiment • Simplified Version L CPS R CPS: Central Photon Source L: Left detector R: Right detector Bell's Theorem

11. Bell’s Gedankenexperiment • The photon has an initial state in the central photon source. • Bell State: • The photon is then shot out to the detectors that will change their state. Bell's Theorem

12. Unitary Operators • The state of the photon is changed by Unitary Operators: • U  and U  • Idea: the Central Photon Source will generate the entangled photons prior to observation. Then the photon will go through the two devices to change their state. Bell's Theorem

13. Bell’s Gedankenexperiment • Full Version: A C B D Bell's Theorem

14. Unitary Operators cos(  ) -sin(  ) sin (  ) cos (  ) U = -sin(  ) cos(  ) -cos(  ) –sin(  ) U = By applying the tensor product of these unitary operators and multiplying it times | we come up with the equation. Bell's Theorem

15. Experimental Fact P( L = R ) = sin2(  -  ) P( L = -R ) = cos2(  -  ) These two equations are derived from this equation. Bell's Theorem

16. Bell’s Gedankenexperiment = [ -sin(+) |00 -cos(+) |01 +cos(+) |10 -sin(+) |11] / Bell's Theorem

17. The probabilities | 00 > = sin2(+) / 2 | 01 > = cos2(+) / 2 | 10 > = cos2(+) / 2 | 11 > = sin2(+) / 2 Bell's Theorem

18. Bell’s Gedankenexperiment The experiment consists of having numerous pairs of entangled photons, one pair after the other, emittedfrom the central source. The left-hand photon of each such pair is randomly forced through either detectorA or detector B, and the right-hand photon is randomly forced through either detector C or detector D. Bell's Theorem

19. Bell’s Gedankenexperiment • Full Version: A C | = |00+|11 2 B D Bell's Theorem

20. Bell’s Gedankenexperiment • Full Version: • Bell’s Tables: • Table 1: Bell's Theorem

21. Bell’s Gedankenexperiment • Full Version: • Bell’s Tables: • Table 2: Bell's Theorem

22. The Theoretical Prediction of K • K is the average of the values of all the plus and minus ones from Table Two. Bell's Theorem

23. Finding Bell’s K • Find the probability that AC = +1 This will be the same as P(A=C) P(A=C)=sin2(67.5° - 135°) =sin2(-67.5°) = sin2(67.5°) • Now since P(AC=+1) is sin2(67.5°) P(AC=-1) is [1- sin2(67.5°) ] = cos2(67.5°) Bell's Theorem

24. Value of all numerical entries in AC is approximately (+1)sin2 (67.5°) + (-1)cos2 (67.5°) = -cos (135°) = Recall that cos2x – sin2x = cos2x Finding Bell’s K Bell's Theorem

25. Being 4 different 2-detector combinations, about ¼ of all entries in AC will be numeric. Thus the sum of numerical entries of the AC column is approximately Similarly treating the other 3 tables and taking the –BD into account, the sum of all numerical entries of Table 2 is approximately Finding Bell’s K Bell's Theorem

26. Found Bell’s K • Table 2 has M rows thus Bell's Theorem

27. Local Reality and Hidden Variable • Local Hidden Variables • Three parts to local hidden variables: • Existence • Locality • Hidden Bell's Theorem

28. Local Reality and Hidden Variable • “Local Hidden Variables: “ • There would be variables that exist whose knowledge would predict correct outcomes of the experiment. • Thus, there should exist two tables, 1’ and 2’, such that all the values in these tables would be complete. Bell's Theorem

29. Bell’s Gedankenexperiment • Complete Knowledge Tables • Table 1’ Bell's Theorem

30. Bell’s Gedankenexperiment • Complete Knowledge Tables • Table 2’ Bell's Theorem

31. Bell’s Theorem • Table 1 and 2 are random samples of 1’ and 2’. They should be the same for the sum of (AC) ~ 1/4 the sum of (AC’). • The distribution of 1’s and -1’s of Table 2 should be the same for 1’s and -1’s of Table 2’. Bell's Theorem

32. Bell’s Theorem of S • S = Grand Sum of Table 2 data • S’ = Grand Sum of Table 2’ Data • K ~ mean of Table 2 • K’ ~ also mean of Table 2’ Bell's Theorem

33. Bell’s Theorem of S • Since S’~4S, K’=K Bell's Theorem

34. Bell’s Theorem of S • Notes for • ith row in table 2’: AC + AD +BC - BD which = A(C+D) + B(C-D) • Suppose C=D, then • Suppose C=-D, then Bell's Theorem

35. Bell’s Theorem of S • k Where • So.. Bell's Theorem

36. The Law of Large Numbers • The more entries in the table, the closer the average comes to K • k • K ~ K’ -> Law of large numbers states K’ becomes closer to K as the entries increase. Bell's Theorem

37. Conclusion • Postulates of Quantum Mechanics • Simplified Version of Bell’s Gedankenexperiment • Full Version of Bell’s Gedankenexperiment • Tables 1 and 2 • Theoretical prediction of K • Tables 1’ and 2’ • Bell’s Contradiction of Table 2’ K’ Value Bell's Theorem

38. Conclusion • Bell’s Gedankenexperiment shows that |K’| should be less than or equal to ½. • It also shows that the value of K’ should be approximately equal to the value K, which is • Therefore, table 2’ cannot exist, thus contradicting that local reality exist. Rather, explained by spooky action at a distance. Bell's Theorem