Entanglement and Bell’s Inequalities Aaron Michalko Kyle Coapman Alberto Sepulveda James MacNeil Madhu Ashok Brian Sheffler
Dec 30, 2015
Entanglement and Bell’s Inequalities
Aaron MichalkoKyle Coapman
Alberto SepulvedaJames MacNeilMadhu AshokBrian 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
Lab Activity 2
• We only changed one polarizer angle this time• What do you think will happen?
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.
•
• No statistical assumptions necessary! • Let’s try it!
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N(A,B ) +N(B,C ) ≥ N(A,C )
CHSH Bell’s Inequality
• Let’s define a measure of correlation E:
• If E=1, perfect correlation. • If E=-1, perfect anticorrelation.
, VV HH VH HVE P P P P
€
EQM (α ,β) = cos2(α − β)
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E HVT (α ,β) =1−β −α
45
, , , ,
,, , , ,
N N N NE
N N N N
Hidden Variable Theory
• Deterministic– Assumes Polarization always has a definite value
that is controlled by a variable– We’ll call the variable λ
(HVT) 1 45,
0, otherwiseVP
HVT v. QM
• Comparing PVV for HVT and QM looks like:
• The look pretty close…but HVT is linear
(HVT) 1,
2 180VVP
(QM) 21, cos
2VVP
CHSH Bell’s Inequality cont.
• Let’s introduce a second measure of correlation:
• According to HVT S≤2 for any angle.
, , ' ', ', 'S E a b E a b E a b E a b
, , , ,
,, , , ,
N N N NE
N N N N
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. Reality2. Locality
Thank You George!!!