Bell’s Theorem and the Demise of Local Reality
Natalia Parshina
Peter Johnson
Josh Robertson
Denise Nagel
James Hardwick
Andy Styve
5/7/2004
Bell's Theorem
1
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
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
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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Bell's Theorem
4
Quantum system can be modeled
by a complex inner product space:
v=Cn
S'
K'=
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4m
Bell's Theorem
5
Evolution of quantum states are
described by unitary operators.
Example: A-1=AT
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Bell's Theorem
6
Quantum measurements are
“described” by a finite set of
projections acting on the state
space being measured.
Suppose the state of a system is:
| ψ > prior to observation, then
P(m) = < ψ | Pm | ψ >
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Bell's Theorem
7
Continued..
If result m occurs, the new state of
the system will be given by:
Pm | ψ >
Pm | ψ >
=
P ( m)
< ψ | Pm | ψ >
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Bell's Theorem
8
The state of a composite (multiparticle) quantum system formed
from:
| ψ 1 >, | ψ 2 >, | ψ 3 >,..., | ψn >
is
| ψ 1 > ⊗ | ψ 2 > ⊗ | ψ 3 > ⊗...⊗ | ψn >
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Bell's Theorem
9
Simplified Version
L
CPS
R
CPS: Central Photon Source
L: Left detector
R: Right detector
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Bell's Theorem
10
The photon has an initial state in
the central photon source.
Bell State:
| 00>+|11> 1(11()1)
1 / 2 * (| 0 > + | 1 >)
ψ=
1(1)
1(1−(−
1)1)
2
1(−1)
The photon is then shot out to the
detectors that will change their
state.
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Bell's Theorem
11
!
The state of the photon is changed
by Unitary Operators:
U λ and U ρ
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 Theorem
12
Full Version:
A
C
| 00>+|11>
ψ=
2
B
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D
Bell's Theorem
13
!
Uλ =
cos( λ )
sin ( λ )
-sin( λ )
cos ( λ )
Uρ =
-sin( ρ ) cos( ρ )
-cos( ρ ) –sin( ρ )
By applying the tensor product of these unitary operators
and multiplying it times |ψ> we come up with the
equation. | ~ > = ( λ ⊗ ρ ) | ψ >
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Bell's Theorem
14
"
P( L = R ) = sin2( λ - ρ )
P( L = -R ) = cos2( λP =- 1ρ1 ⊗) 1 1
11
These two Pequations
(L = R ) = ψ~are
P ψ~derived
+ ψ~ P ψ~
from this equation. | ~ > = ( λ ⊗
00
P 00 = 0 0 ⊗ 0 0
11
ρ
) |ψ >
P11 = 1 1 ⊗ 1 1
P(L = R ) = ψ~ P 00 ψ~ + ψ~ P11ψ~
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Bell's Theorem
15
| ψ~ >
| ψ~ > = [ -sin(λ+ρ)
| ψ~ > |00>
-cos(λ+ρ) |01>
+cos(λ+ρ) |10>
-sin(λ+ρ) |11>] / 2
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Bell's Theorem
16
#
| 00 >
| 01 >
| 10 >
| 11 >
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=
=
=
=
$ $
sin2(λ+ρ) / 2
cos2(λ+ρ) / 2
cos2(λ+ρ) / 2
sin2(λ+ρ) / 2
Bell's Theorem
17
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
Full Version:
A
C
|ϕ> = |00>+|11>
√2
B
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D
Bell's Theorem
19
Full Version:
Bell’s Tables:
•
Table 1:
A
1
?
.
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Bell's Theorem
B
?
- 1
.
C
?
?
.
D
- 1
- 1
.
20
Full Version:
Bell’s Tables:
•
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Table 2:
AC AD
? - 1
BC
?
- BD
?
?
.
?
.
- 1
.
Bell's Theorem
?
.
21
#
#
%
K is the average of the values
of all the plus and minus ones
from Table Two.
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Bell's Theorem
22
"
&
%
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°)
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Bell's Theorem
23
"
&
%
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
"
&
%
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
2
2
2
M 2
[
+
+
+
]
4 2
2
2
2
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Bell's Theorem
25
"
%
Table 2 has M rows thus
2
K≈
2
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Bell's Theorem
26
'
)
(
*
$
Local Hidden Variables
Three parts to local hidden
variables:
Existence
Locality
Hidden
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Bell's Theorem
27
'
)
(
*
$
“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
Complete Knowledge Tables
Table 1’
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A
a1
B
b1
C
c1
D
d1
a2
a3
..
b2
b3
c2
c3
d2
d3
..
..
..
Bell's Theorem
29
Complete Knowledge Tables
Table 2’
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AC
AD
BC
- BD
ac1
ad1
bc1
- bd1
ac2
ad2
bc2
- bd2
ac3
ad3
bc3
- bd3
..
..
..
..
Bell's Theorem
30
#
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’.
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4
( AC ) ≅
( AC ' ) 4
( BC ) ≅
4
( AD) ≅
( AD' ) 4
(− BD ) ≅
Bell's Theorem
( BC ' )
(− BD ' )
31
#
S = Grand Sum of Table 2 data
S’ = Grand Sum of Table 2’ Data
S ' =AC + AC
AD +'+
≈ 4( AC +
≈ 4(
AD−'BD
+ ) = 4BC
'+
S
AD + BC +
BC +
− BD'
− BD ) = 4 S
S ' ≈ 4S
K ~ mean of Table 2
K’ ~ also mean of Table 2’
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Bell's Theorem
32
#
S'
K'=
4m
S
K=
m
Since S’~4S, K’=K
4S S
= ≈K
4m m
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Bell's Theorem
33
#
1
Notes for K ' ≤ 2
ith row in table 2’: AC + AD +BC- BD
which =1 A(C+D) + B(C
- D)
K '≤ + ±/ −2 2
2 AC
2 AC
2
±2
Suppose
C=D, then 2 AC
Suppose C=
- D,then 2 AC
±2
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Bell's Theorem
34
#
− 2m ≤ S ' ≤ 2m
S'
1 S' 1
= K'
−k ≤
≤ Where
2 4m 2
4m
1
So.. K ' ≤
2
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Bell's Theorem
35
#
,
' +
$
'
&
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.
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Bell's Theorem
36
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
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Bell's Theorem
37
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.
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Bell's Theorem
38