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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
04/18/23 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.
04/18/23 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.
04/18/23 Bell's Theorem 5
Postulates of Quantum Mechanics Quantum system can be modeled
by a complex inner product space: v=Cn
m
SK
4
''
04/18/23 Bell's Theorem 6
Postulates of Quantum Mechanics Evolution of quantum states are
described by unitary operators. Example: A-1=AT
04/18/23 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) =
| || mP
04/18/23 Bell's Theorem 8
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
04/18/23 Bell's Theorem 9
Postulates of Quantum Mechanics The state of a composite (multi-
particle) quantum system formed from:
is n |,...,|,|,| 321
n |...||| 321
04/18/23 Bell's Theorem 10
Bell’s Gedankenexperiment Simplified Version
CPSL R
CPS: Central Photon Source
L: Left detector
R: Right detector
04/18/23 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.
)1|0(|*2/1
)1(1
)1(1
)1(1
)1(1
)1(1
)1(1
2
11|00|
04/18/23 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.
04/18/23 Bell's Theorem 14
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
04/18/23 Bell's Theorem 15
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
04/18/23 Bell's Theorem 16
Bell’s Gedankenexperiment
= [ -sin(+) |00
-cos(+) |01+cos(+) |10
-sin(+) |11] /
~|
~| ~|
2
04/18/23 Bell's Theorem 17
The probabilities
| 00 > = sin2(+) / 2| 01 > = cos2(+) / 2| 10 > = cos2(+) / 2| 11 > = sin2(+) / 2
04/18/23 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.
04/18/23 Bell's Theorem 20
Bell’s Gedankenexperiment
Full Version: Bell’s Tables:
• Table 1:
A B C D
1 ? ? -1
? -1 ? -1
. . . .
04/18/23 Bell's Theorem 21
Bell’s Gedankenexperiment
Full Version: Bell’s Tables:
• Table 2:
AC AD BC -BD
? -1 ? ?
? ? ? -1
. . . .
04/18/23 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.
04/18/23 Bell's Theorem 23
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
04/18/23 Bell's Theorem 24
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
04/18/23 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
2
2
4
M
]2
2
2
2
2
2
2
2[
4
M
Finding Bell’s K
04/18/23 Bell's Theorem 27
Local Reality and Hidden Variable Local Hidden Variables
Three parts to local hidden variables:
Existence Locality Hidden
04/18/23 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.
04/18/23 Bell's Theorem 29
Bell’s Gedankenexperiment Complete Knowledge Tables
Table 1’
A B C D
a1 b1 c1 d1
a2 b2 c2 d2
a3 b3 c3 d3
.. .. .. ..
04/18/23 Bell's Theorem 30
Bell’s Gedankenexperiment Complete Knowledge Tables
Table 2’
AC AD BC -BD
ac1 ad1 bc1 -bd1
ac2 ad2 bc2 -bd2
ac3 ad3 bc3 -bd3
.. .. .. ..
04/18/23 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’.
)'()(4 ACAC )'()(4 BCBC
)'()(4 ADAD )'()(4 BDBD
04/18/23 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’
SS 4'
SBDBCADAC 44 ''''' BDBCADACS
SBDBCADAC 44
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
04/18/23 Bell's Theorem 35
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