Transcript Bell’s Inequality

Bell’s Inequality
Quantum Mechanics(14/2)
Hoseong Lee
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!”
Quantum Mechanics(14/2)
Hoseong Lee
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!”
Quantum Mechanics(14/2)
Hoseong Lee
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
Quantum Mechanics(14/2)
Hoseong Lee
Bell’s inequality
• Simple version of a bell’s inequality
– Various possible polarization combinations of the two EPR photons
Quantum Mechanics(14/2)
Hoseong Lee
Bell’s inequality
• Simple version of a bell’s inequality
– 𝑅1 ⊂ 𝑅2 ∪ 𝑅3
⇒ 𝑃 𝑅1 ≤ 𝑃 𝑅2 + 𝑃 𝑅3
– Specific example for a Bell’s inequality.
Quantum Mechanics(14/2)
Hoseong Lee
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)
Quantum Mechanics(14/2)
Hoseong Lee
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
Quantum Mechanics(14/2)
Hoseong Lee
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
Quantum Mechanics(14/2)
Hoseong Lee
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.
Quantum Mechanics(14/2)
Hoseong Lee
Bell’s inequality
• Simple version of a bell’s inequality
– Quantum mechanical calculation
• 0.2500 > 0.0732 + 0.1464
• A calculation that also appears to agree well with experiment.
Quantum Mechanics(14/2)
Hoseong Lee
Bell’s inequality experiments
• Notable experiments
– Freedman and Clauser, 1972
• First actual Bell test
– Aspect, 1981-2
• 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
Quantum Mechanics(14/2)
Hoseong Lee
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
Quantum Mechanics(14/2)
Hoseong Lee
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”)
Quantum Mechanics(14/2)
Hoseong Lee
Bell’s inequality experiments
• CHSH experiment
– The experimental estimate for E(a,b) is then calculated as:
𝑁++ − 𝑁+− − 𝑁−+ + 𝑁−−
𝐸=
𝑁++ + 𝑁+− + 𝑁−+ + 𝑁−−
– 𝑆𝑒𝑥𝑝𝑡 = 2.697 ± 0.015 > 2
– 𝑆𝑄𝑀 = 2.70 ± 0.05 > 2
– Demonstration to non-locality of quantum mechanics
Quantum Mechanics(14/2)
Hoseong Lee