6/9/2015Bell's Theorem1 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

6/9/2015Bell's Theorem2 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

6/9/2015Bell's Theorem3 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.

6/9/2015Bell's Theorem4 Postulates of Quantum Mechanics  Quantum system can be modeled by a complex inner product space: v = C n  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 X 1, X 2, …,X n is the tensor product of the set.

6/9/2015Bell's Theorem5 Postulates of Quantum Mechanics Quantum system can be modeled by a complex inner product space: v=C n

6/9/2015Bell's Theorem6 Postulates of Quantum Mechanics Evolution of quantum states are described by unitary operators. Example: A -1 =A T

6/9/2015Bell's Theorem7 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) =

6/9/2015Bell's Theorem8 Postulates of Quantum Mechanics Continued.. If result m occurs, the new state of the system will be given by:

6/9/2015Bell's Theorem9 Postulates of Quantum Mechanics The state of a composite (multi- particle) quantum system formed from: is

6/9/2015Bell's Theorem10 Bell’s Gedankenexperiment Simplified Version CPSLR CPS: Central Photon Source L: Left detector R: Right detector

6/9/2015Bell's Theorem11 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.

6/9/2015Bell's Theorem12 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.

6/9/2015Bell's Theorem13 Bell’s Gedankenexperiment Full Version: A D C B

6/9/2015Bell's Theorem14 U  = -sin(  ) cos(  ) -cos(  ) –sin(  ) cos( ) -sin( ) sin ( ) cos ( ) U = Unitary Operators By applying the tensor product of these unitary operators and multiplying it times |  we come up with the equation.

6/9/2015Bell's Theorem15 Experimental Fact P( L = R ) = sin 2 ( -  ) P( L = -R ) = cos 2 ( -  ) These two equations are derived from this equation.

6/9/2015Bell's Theorem16 Bell’s Gedankenexperiment = [ -sin( +  ) |00  -cos( +  ) |01  +cos( +  ) |10  -sin( +  ) |11  ] /

6/9/2015Bell's Theorem17 The probabilities | 00 > = sin 2 ( +  ) / 2 | 01 > = cos 2 ( +  ) / 2 | 10 > = cos 2 ( +  ) / 2 | 11 > = sin 2 ( +  ) / 2

6/9/2015Bell's Theorem18 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.

6/9/2015Bell's Theorem19 Bell’s Gedankenexperiment Full Version: |  = |00  +|11  A D C B 22

6/9/2015Bell's Theorem20 Bell’s Gedankenexperiment Full Version:  Bell’s Tables: Table 1: ABCD 1?? ? ?....

6/9/2015Bell's Theorem21 Bell’s Gedankenexperiment Full Version:  Bell’s Tables: Table 2: ACADBC-BD ??? ???....

6/9/2015Bell's Theorem22 The Theoretical Prediction of K K is the average of the values of all the plus and minus ones from Table Two.

6/9/2015Bell's Theorem23 Find the probability that AC = +1 This will be the same as P(A=C) P(A=C)=sin 2 (67.5° - 135°) =sin 2 (-67.5°) = sin 2 (67.5°) Now since P(AC=+1) is sin 2 (67.5°) P(AC=-1) is [1- sin 2 (67.5°) ] = cos 2 (67.5°) Finding Bell’s K

6/9/2015Bell's Theorem24 Value of all numerical entries in AC is approximately (+1)sin 2 (67.5°) + (-1)cos 2 (67.5°) =-cos (135°) = Recall that cos 2 x – sin 2 x = cos2x Finding Bell’s K

6/9/2015Bell's Theorem25 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

6/9/2015Bell's Theorem26 Table 2 has M rows thus Found Bell’s K

6/9/2015Bell's Theorem27 Local Reality and Hidden Variable Local Hidden Variables  Three parts to local hidden variables:  Existence  Locality  Hidden

6/9/2015Bell's Theorem28 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.

6/9/2015Bell's Theorem29 Bell’s Gedankenexperiment Complete Knowledge Tables  Table 1’ ABCD a1a1 b1b1 c1c1 d1d1 a2a2 b2b2 c2c2 d2d2 a3a3 b3b3 c3c3 d3d3..

6/9/2015Bell's Theorem30 Bell’s Gedankenexperiment Complete Knowledge Tables  Table 2’ ACADBC-BD ac 1 ad 1 bc 1 -bd 1 ac 2 ad 2 bc 2 -bd 2 ac 3 ad 3 bc 3 -bd 3..

6/9/2015Bell's Theorem31 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’.

6/9/2015Bell's Theorem32 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’

6/9/2015Bell's Theorem33 Bell’s Theorem of S Since S’~4S, K’=K

6/9/2015Bell's Theorem34 Bell’s Theorem of S Notes for  i th row in table 2’: AC + AD +BC - BD which = A(C+D) + B(C-D)  Suppose C=D, then  Suppose C=-D, then

6/9/2015Bell's Theorem35 Bell’s Theorem of S k Where So..

6/9/2015Bell's Theorem36 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.

6/9/2015Bell's Theorem37 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

6/9/2015Bell's Theorem38 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.