Entanglement and Bell’s
Inequalities
Aaron Michalko
Kyle Coapman
Alberto Sepulveda
James MacNeil
Madhu Ashok
Brian Sheffler
Correlation
• Drawer of Socks
– 2 colors, Red and Blue,
– Four combinations: RR, RB, BR, BB
– (pR1 + qB1) (pR2 + qB2)
– 50% Same, 50% Different
– NO CORRELATION
Correlation
•
•
•
•
What if socks are paired: RR, BB
If you know one, you know the other
100% Same, 0% Different
Perfectly Correlated
• Entanglement ~ Correlation
What is Entanglement?
• Correlation in all bases
• What is a basis?
– Like a set of axes
– Our basis is polarization: V and H
– Photons either VV or HH
– Perfectly correlated
How do we Entangle Photons?
• Parametric down conversion
– Non-linear, birefringent crystal
– 2 emitted photons, signal and idler
How do we Entangle Photons?
• 2 crystals create overlapping cones of photons
• Photons are entangled:
– We don’t know if any photon is VV or HH…or
maybe both…
Logic Exercise
• Three Assumptions:
– When a photon leaves the source it is either H or
V
– No communication between photons after
emission
– Nothing that we don’t know, V/H is a complete
description
Logic Exercise
• Polarizers set at 45
• 50% transmit at each polarizer
• Logical Conclusion:
– 25% Coincident
– 50% One at a time
– 25% No Detection
>>> NO CORRELATION
Logic Exercise
•
•
•
•
Entangled Source
50% coincidence reading
50% no reading
>>>100% Correlation
Lab setup
Lab setup
Lab Activity 1
• We measured the coincidence counts of
entangled photons
• Each passed through a polarizer set at the
same angle
Number of Counts
Count vs. Angle of Alpha
80000
70000
60000
50000
40000
30000
20000
10000
0
Count A Net
Count B Net
0
50
100
150
200
250
Angle of Alpha
300
350
400
Lab Activity 2
• We only changed one polarizer angle this time
• What do you think will happen?
Changing Alpha
800
Number of Counts
700
600
Count (A = 0)
500
Count (A = 90)
400
Count (A = 45)
300
Count (A = 135)
200
100
0
0
50
100
150
200
Polarizer angle
250
300
350
400
Logic Exercise
• Which assumption is incorrect:
– Reality
– Locality
– Hidden Variables
Bell’s Inequalities
• Let A,B and C be three binary characteristics.
• Assumptions: Logic is valid. The parameters
exist whether they are measured or not.
•
N(A,B )  N(B,C )  N(A,C )
• No statistical assumptions necessary!
• Let’s try it!
CHSH Bell’s Inequality
• Let’s define a measure of correlation E:
E  ,    PVV  PHH  PVH  PHV
E  ,   
E
QM
N  ,    N   ,     N  ,     N   ,  
N  ,    N   ,     N  ,     N   ,  
(, )  cos (  )
2
E
HVT
(  , )  1 
• If E=1, perfect correlation.
• If E=-1, perfect anticorrelation.


45
Hidden Variable Theory
• Deterministic
– Assumes Polarization always has a definite value
that is controlled by a variable
– We’ll call the variable λ
(HVT)
V
P
1
 ,    
0,
    45
otherwise
HVT v. QM
• Comparing PVV for HVT and QM looks like:
(HVT)
VV
P
(QM)
VV
P
1  
 ,    
2
180
1
2

,


cos
 
  
2
• The look pretty close…but HVT is linear
CHSH Bell’s Inequality cont.
• Let’s introduce a second measure of
correlation:
S  E  a, b   E  a, b '  E  a ', b   E  a ', b ' 
E  ,   
N  ,    N   ,     N  ,     N   ,  
N  ,    N   ,     N  ,     N   ,  
• According to HVT S≤2 for any angle.
CHSH Bell’s Inequality cont.
•
•
•
•
QM predicts S≥2 in some cases.
a=-45°, a’=0°, b=22.5°, b’=-22.5°
S(QM)=2.828
S(HVT)=2
This means that either locality or reality are
false assumptions!
Our Lab Activity
• We recorded coincidence counts with
combinations of | polarization angles
• S = 2.25
• We violated Bell’s inequality! That means our
system is inherently quantum, and cannot be
explained using classical physics
This is a little scary…
• HVT is not a valid explanation for the behavior
of entangled photons
• So…that means we either violate:
1. Reality
2. Locality
Thank You George!!!