Bell`s Theorem

```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

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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.
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Bell's Theorem
3
Postulates of Quantum
Mechanics




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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'
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4m
Bell's Theorem
5
Postulates of Quantum
Mechanics


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Evolution of quantum states are
described by unitary operators.
Example: A-1=AT
Bell's Theorem
6
Postulates of Quantum
Mechanics


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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)
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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 
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Bell's Theorem
9
Bell’s
Gedankenexperiment

L
Simplified Version
CPS
R
CPS: Central Photon Source
L: Left detector
R: Right detector
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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

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 1(1) 
The photon is then shot out to the
detectors that will change their
state.
Bell's Theorem
11
Unitary Operators



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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
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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~   (    ) |  
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Bell's Theorem
14
Experimental Fact
P( L = R ) = sin2(  -  )
P( L = -R ) = cos2( P - 11 ) 1 1
11
These two equations
derived
PL  R  ~are
P ~
 ~ P ~
from this equation. | ψ~   (    ) |  
00
11
P0 0  0 0  0 0 P1 1  1 1  1 1
PL  R  ~ P0 0 ~  ~ P1 1~
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Bell's Theorem
15
Bell’s
Gedankenexperiment
| ~ 
| ~  = [ -sin(+)
| ~  |00
-cos(+) |01
+cos(+) |10
-sin(+) |11] / 2
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Bell's Theorem
16
The probabilities
| 00 >
| 01 >
| 10 >
| 11 >
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=
=
=
=
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.
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Bell's Theorem
18
Bell’s
Gedankenexperiment

Full Version:
A
C
| = |00+|11
2
B
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D
Bell's Theorem
19
Bell’s
Gedankenexperiment

Full Version:

Bell’s Tables:
•
Table 1:
A
1
?
.
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Bell's Theorem
B
?
-1
.
C
?
?
.
D
-1
-1
.
20
Bell’s
Gedankenexperiment

Full Version:

Bell’s Tables:
•
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Table 2:
? -1
BC
?
-BD
?
?
.
?
.
-1
.
Bell's Theorem
?
.
21
The Theoretical
Prediction of K

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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


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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°)
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
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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
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Bell's Theorem
25
Found Bell’s K

Table 2 has M rows thus
2
K
2
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Bell's Theorem
26
Local Reality and
Hidden Variable

Local Hidden Variables

Three parts to local hidden
variables:
 Existence
 Locality
 Hidden
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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.

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Bell's Theorem
28
Bell’s
Gedankenexperiment

Complete Knowledge Tables

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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

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Table 2’
AC
BC
-BD
ac1
bc1
-bd1
ac2
bc2
-bd2
ac3
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' )
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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


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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
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Bell's Theorem
33
Bell’s Theorem of S

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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

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1
So.. K ' 
2
Bell's Theorem
35
The Law of Large
Numbers



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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







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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



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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,