1. Bell’s Inequality

2. Hidden variables Hidden variable theory
Argument about uncertainty property of quantum mechanics
Hidden variable
Investing quantum mechanics with local realism
Underlying deterministic unknown variable in quantum mechanics
Bohm’s hidden variable theory
“God does not play dice!”

3. Hidden variables Local hidden variable theory Locality
Principle that an object is only directly influenced by its immediate surroundings
EPR paradox – showed non-locality of quantum mechanics
Two photon that separated so far apart
The measurement of one photon ⇒ determining the other one’s states
Local hidden variable
a quantity whose value is presently unknown with local property
“Spooky action at a distance!”

4. Bell’s inequality Bell’s theorem
Proposed by John Stewart Bell, in the paper that “On the Einstein-Podolsky-Rosen paradox”, 1964.
A way of distinguishing experimentally between local hidden variable theories and the predictions of quantum mechanics
Bell’s inequality →
Inequality that derived from local hidden variable theory
Any quantum correlations under local hidden variable theory do not satisfy bell’s inequalities.
Demonstration by bell test experiments

5. Bell’s inequality Simple version of a bell’s inequality
Various possible polarization combinations of the two EPR photons

6. Bell’s inequality Simple version of a bell’s inequality
𝑅 1 ⊂ 𝑅 2 ∪ 𝑅 3 ⇒ 𝑃 𝑅 1 ≤𝑃 𝑅 2 +𝑃 𝑅 3
Specific example for a Bell’s inequality.

7. Bell’s inequality Simple version of a bell’s inequality
Quantum mechanical calculation
Photon linearly polarized at an angle θ to the horizontal
Probability that such a photon will pass a horizontally-oriented beamsplitter
Two different photon modes: propagating to the left (L) and to the right (R)
EPR state with the two orthogonal polarization states (generalized form: θ and θ+π/2, instead of horizontal and vertical)

8. Bell’s inequality Simple version of a bell’s inequality
Quantum mechanical calculation
Consider that examine this state with a horizontal polarizer on the left and a polarizer at angle φ to the horizontal on the right, then amplitude is
Note that we can write

9. Bell’s inequality Simple version of a bell’s inequality
Quantum mechanical calculation
Amplitude is independent of the angle θ of the polarization axis of the EPR pair.
We can conclude that the probability of the “left” photon passing the left polarizer at angle 0 and the “right” photon passing the right polarizer at angle φ is

10. Bell’s inequality Simple version of a bell’s inequality
Quantum mechanical calculation
If a photon on the right passes at an angle φ, then it fails to emerge from the the other arm of a polarization beamsplitter, an arm that passes a photon of polarization angle φ-π/2.
Probability of the “left” photon passing the left polarizer at angle 0 and the “right” photon failing to pass the right polarizer at angle φ is
The choice of the polarizer orientation on the left as “horizontal” is arbitrary.

11. Bell’s inequality Simple version of a bell’s inequality
Quantum mechanical calculation
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A calculation that also appears to agree well with experiment.

12. Bell’s inequality experiments
Notable experiments
Freedman and Clauser, 1972
First actual Bell test
Aspect,
First and last used the CH74 inequality, 1981
First application of the CHSH inequality, 1982
Tittel and the Geneva group, 1998
Long distance of several kilometers
Salart et al., 2008
Long distance of 18 km

13. Bell’s inequality experiments
CHSH experiment
Proposed by John Clauser, Michael Horne, Abner Shimony, and Richard Holt, in the paper that “Bell’s theorem; experimental tests and implications”, 1969.
CHSH inequality
Quantum mechanics calculation: 𝑆≤2 2
CHSH violations predicted by the theory of quantum mechanics

14. Bell’s inequality experiments
CHSH experiment
Set of four correlations; { ‘++’, ‘+-’, ‘-+’, ‘--’ }
Polarization: vertical (V or +) and horizontal (H or -)
Coincidence counts: { N++, N--, N+-, N-+ }
{ a, a’, b, b’ } ⇒ { 0˚, 45˚, 22.5˚, 67.5˚ } (‘Bell test angles”)

15. Bell’s inequality experiments
CHSH experiment
The experimental estimate for E(a,b) is then calculated as:
𝑆 𝑒𝑥𝑝𝑡 =2.697±0.015>2
𝑆 𝑄𝑀 =2.70±0.05>2
Demonstration to non-locality of quantum mechanics
𝐸= 𝑁 ++ − 𝑁 +− − 𝑁 −+ + 𝑁 −− 𝑁 ++ + 𝑁 +− + 𝑁 −+ + 𝑁 −−