Report

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 4/8/2015 Bell's Theorem 1 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 4/8/2015 Bell's Theorem 2 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. 4/8/2015 Bell's Theorem 3 Postulates of Quantum Mechanics 4/8/2015 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 4 Postulates of Quantum Mechanics Quantum system can be modeled by a complex inner product space: v=Cn S' K' 4/8/2015 4m Bell's Theorem 5 Postulates of Quantum Mechanics 4/8/2015 Evolution of quantum states are described by unitary operators. Example: A-1=AT Bell's Theorem 6 Postulates of Quantum Mechanics 4/8/2015 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) = | Pm | Bell's Theorem 7 Postulates of Quantum Mechanics Continued.. If result m occurs, the new state of the system will be given by: Pm | Pm | | Pm | P(m) 4/8/2015 Bell's Theorem 8 Postulates of Quantum Mechanics The state of a composite (multiparticle) quantum system formed from: | 1 , | 2 , | 3 ,...,| n is | 1 | 2 | 3 ... | n 4/8/2015 Bell's Theorem 9 Bell’s Gedankenexperiment L Simplified Version CPS R CPS: Central Photon Source L: Left detector R: Right detector 4/8/2015 Bell's Theorem 10 Bell’s Gedankenexperiment The photon has an initial state in the central photon source. Bell State: | 00 | 11 1(11()1) 1/ 2 * (| 0 | 1 ) 1(1) 1(1( 1)1) 2 4/8/2015 1(1) The photon is then shot out to the detectors that will change their state. Bell's Theorem 11 Unitary Operators 4/8/2015 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 12 Bell’s Gedankenexperiment Full Version: A C | 00 | 11 2 B 4/8/2015 D Bell's Theorem 13 Unitary Operators U = cos( ) sin ( ) -sin( ) cos ( ) U = -sin( ) cos( ) -cos( ) –sin( ) By applying the tensor product of these unitary operators and multiplying it times | we come up with the equation. | ψ~ ( ) | 4/8/2015 Bell's Theorem 14 Experimental Fact P( L = R ) = sin2( - ) P( L = -R ) = cos2( P - 11 ) 1 1 11 These two equations derived PL R ~are P ~ ~ P ~ from this equation. | ψ~ ( ) | 00 11 P0 0 0 0 0 0 P1 1 1 1 1 1 PL R ~ P0 0 ~ ~ P1 1~ 4/8/2015 Bell's Theorem 15 Bell’s Gedankenexperiment | ~ | ~ = [ -sin(+) | ~ |00 -cos(+) |01 +cos(+) |10 -sin(+) |11] / 2 4/8/2015 Bell's Theorem 16 The probabilities | 00 > | 01 > | 10 > | 11 > 4/8/2015 = = = = sin2(+) / 2 cos2(+) / 2 cos2(+) / 2 sin2(+) / 2 Bell's Theorem 17 Bell’s Gedankenexperiment The experiment consists of having numerous pairs of entangled photons, one pair after the other, emitted from the central source. The left-hand photon of each such pair is randomly forced through either detector A or detector B, and the right-hand photon is randomly forced through either detector C or detector D. 4/8/2015 Bell's Theorem 18 Bell’s Gedankenexperiment Full Version: A C | = |00+|11 2 B 4/8/2015 D Bell's Theorem 19 Bell’s Gedankenexperiment Full Version: Bell’s Tables: • Table 1: A 1 ? . 4/8/2015 Bell's Theorem B ? -1 . C ? ? . D -1 -1 . 20 Bell’s Gedankenexperiment Full Version: Bell’s Tables: • 4/8/2015 Table 2: AC AD ? -1 BC ? -BD ? ? . ? . -1 . Bell's Theorem ? . 21 The Theoretical Prediction of K 4/8/2015 K is the average of the values of all the plus and minus ones from Table Two. Bell's Theorem 22 Finding Bell’s K 4/8/2015 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 23 Finding Bell’s K Recall that cos2x – sin2x = cos2x [ 2 2 2 2 ] 2 2 2 2 Value of all numerical entries in AC is approximately (+1)sin2 (67.5°) + (-1)cos2 (67.5°) 2 = -cos (135°) = 2 4/8/2015 Bell's Theorem 24 Finding Bell’s K 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 M 2 4 2 Similarly treating the other 3 tables and taking the –BD into account, the sum of all numerical entries of Table 2 is approximately M 2 2 2 2 [ ] 4 2 2 2 2 4/8/2015 Bell's Theorem 25 Found Bell’s K Table 2 has M rows thus 2 K 2 4/8/2015 Bell's Theorem 26 Local Reality and Hidden Variable Local Hidden Variables Three parts to local hidden variables: Existence Locality Hidden 4/8/2015 Bell's Theorem 27 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. 4/8/2015 Bell's Theorem 28 Bell’s Gedankenexperiment Complete Knowledge Tables 4/8/2015 Table 1’ A a1 B b1 C c1 D d1 a2 a3 .. b2 b3 c2 c3 d2 d3 .. .. .. Bell's Theorem 29 Bell’s Gedankenexperiment Complete Knowledge Tables 4/8/2015 Table 2’ AC AD BC -BD ac1 ad1 bc1 -bd1 ac2 ad2 bc2 -bd2 ac3 ad3 bc3 -bd3 .. .. .. .. Bell's Theorem 30 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’. 4 ( AC) ( AC' ) 4(BC) (BC' ) 4 ( AD) ( AD' ) 4(BD) (BD' ) 4/8/2015 Bell's Theorem 31 Bell’s Theorem of S S = Grand Sum of Table 2 data S’ = Grand Sum of Table 2’ Data S ' AC' AD' BC' BD' 4 AC AD BC BD 4S 4 AC AD BC BD 4S S ' 4S 4/8/2015 K ~ mean of Table 2 K’ ~ also mean of Table 2’ Bell's Theorem 32 Bell’s Theorem of S S' K' 4m S K m Since S’~4S, K’=K 4S S K 4m m 4/8/2015 Bell's Theorem 33 Bell’s Theorem of S 4/8/2015 1 Notes for K ' 2 ith row in table 2’: AC + AD +BC - BD which =1 A(C+D) + B(C-D) K '/ 2 AC 2 AC 22 2 Suppose C=D, then 2 AC 2 Suppose C=-D, then 2 AC 2 Bell's Theorem 34 Bell’s Theorem of S 2m S ' 2m S' 1 S' 1 K' k Where 2 4m 2 4m 4/8/2015 1 So.. K ' 2 Bell's Theorem 35 The Law of Large Numbers 4/8/2015 The more entries in the table, the closer the average comes to K k K' K KK ~ K’ K ' -> Law of large numbers states K’ becomes closer to K as the entries increase. Bell's Theorem 36 Conclusion 4/8/2015 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 37 Conclusion 4/8/2015 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 2 the value K, which is 2 Therefore, table 2’ cannot exist, thus contradicting that local reality exist. Rather, explained by spooky action at a distance. Bell's Theorem 38