Bell’s Inequality

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!”

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!”

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

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

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.

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)

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

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

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.

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.

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

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

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”)

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 −−