04/18/23 Bell's Theorem 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

04/18/23 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

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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.

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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.

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Postulates of Quantum Mechanics Quantum system can be modeled

by a complex inner product space: v=Cn

m

SK

4

''

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Postulates of Quantum Mechanics Evolution of quantum states are

described by unitary operators. Example: A-1=AT

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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) =

| || mP

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Postulates of Quantum MechanicsContinued..

If result m occurs, the new state of the system will be given by:

)(

|

||

|

mP

P

P

P m

m

m

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Postulates of Quantum Mechanics The state of a composite (multi-

particle) quantum system formed from:

is n |,...,|,|,| 321

n |...||| 321

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Bell’s Gedankenexperiment Simplified Version

CPSL R

CPS: Central Photon Source

L: Left detector

R: Right detector

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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.

)1|0(|*2/1

)1(1

)1(1

)1(1

)1(1

)1(1

)1(1

2

11|00|

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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.

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Bell’s Gedankenexperiment

Full Version:

A

D

C

B

2

11|00|

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U = -sin( ) cos( )-cos( ) –sin( )

cos( ) -sin( )sin ( ) cos ( )U =

Unitary Operators

By applying the tensor product of these unitary operatorsand multiplying it times | we come up with the equation. |) ( ~|

ψ

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Experimental Fact

~~~~1100 PPRLP

P( L = R ) = sin2( - )

P( L = -R ) = cos2( - )

These two equations are derivedfrom this equation. |) ( ~|

ψ

111111 P

000000 P 111111 P

~~~~1100 PPRLP

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Bell’s Gedankenexperiment

= [ -sin(+) |00

-cos(+) |01+cos(+) |10

-sin(+) |11] /

~|

~| ~|

2

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The probabilities

| 00 > = sin2(+) / 2| 01 > = cos2(+) / 2| 10 > = cos2(+) / 2| 11 > = sin2(+) / 2

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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.

04/18/23 Bell's Theorem 19

Bell’s Gedankenexperiment

Full Version:

| = |00+|11

A

D

C

B

2

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Bell’s Gedankenexperiment

Full Version: Bell’s Tables:

• Table 1:

A B C D

1 ? ? -1

? -1 ? -1

. . . .

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Bell’s Gedankenexperiment

Full Version: Bell’s Tables:

• Table 2:

AC AD BC -BD

? -1 ? ?

? ? ? -1

. . . .

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The Theoretical Prediction of K

K is the average of the values of all the plus and minus ones from Table Two.

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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°)

Finding Bell’s K

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Value of all numerical entries in AC is approximately

(+1)sin2 (67.5°) + (-1)cos2 (67.5°)

= -cos (135°) =

]2

2

2

2

2

2

2

2[

2

2

Recall that

cos2x – sin2x = cos2x

Finding Bell’s K

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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

2

2

4

M

]2

2

2

2

2

2

2

2[

4

M

Finding Bell’s K

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Table 2 has M rows thus

2

2K

Found Bell’s K

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Local Reality and Hidden Variable Local Hidden Variables

Three parts to local hidden variables:

Existence Locality Hidden

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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.

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Bell’s Gedankenexperiment Complete Knowledge Tables

Table 1’

A B C D

a1 b1 c1 d1

a2 b2 c2 d2

a3 b3 c3 d3

.. .. .. ..

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Bell’s Gedankenexperiment Complete Knowledge Tables

Table 2’

AC AD BC -BD

ac1 ad1 bc1 -bd1

ac2 ad2 bc2 -bd2

ac3 ad3 bc3 -bd3

.. .. .. ..

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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 ACAC )'()(4 BCBC

)'()(4 ADAD )'()(4 BDBD

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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’

SS 4'

SBDBCADAC 44 ''''' BDBCADACS

SBDBCADAC 44

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Bell’s Theorem of S

Since S’~4S, K’=K

m

SK

4

''

m

SK

Km

S

m

S

4

4

04/18/23 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

2

1'K

2

1' K

2/2 AC 22 AC22 AC

22 AC

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Bell’s Theorem of S

k Where

So..

mSm 2'2

2

1

4

'

2

1

m

S

2

1' K

'4

'K

m

S

04/18/23 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.

'KK

'KK

04/18/23 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

04/18/23 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.

2

2