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Entanglement and Bell’s Inequalities
Aaron MichalkoKyle Coapman
Alberto SepulvedaJames MacNeilMadhu AshokBrian Sheffler
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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
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Correlation
• What if socks are paired: RR, BB• If you know one, you know the other• 100% Same, 0% Different• Perfectly Correlated
• Entanglement ~ Correlation
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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
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How do we Entangle Photons?
• Parametric down conversion– Non-linear, birefringent crystal– 2 emitted photons, signal and idler
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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…
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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
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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
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Logic Exercise
• Entangled Source• 50% coincidence reading• 50% no reading• >>>100% Correlation
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Lab setup
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Lab setup
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Lab Activity 1
• We measured the coincidence counts of entangled photons
• Each passed through a polarizer set at the same angle
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Lab Activity 2
• We only changed one polarizer angle this time• What do you think will happen?
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Logic Exercise
• Which assumption is incorrect:– Reality– Locality– Hidden Variables
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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!
€
N(A,B ) +N(B,C ) ≥ N(A,C )
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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)
€
E HVT (α ,β) =1−β −α
45
, , , ,
,, , , ,
N N N NE
N N N N
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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
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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
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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
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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!
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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
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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
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Thank You George!!!