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

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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!!!