u Before 20 th century Classical Mechanics - Absolute space & - PowerPoint PPT Presentation

u Before 20 th century • Classical Mechanics - Absolute space & time - Matter = particle - Light = wave Young’s double-slit experiment Isaac Thomas Newton Young Particle Wave

u 20 th century • Einstein’s photoelectric effect experiment(1905) - Duality of light • De Broglie’s matter wave(1924) - Duality of matter • Stern-Gerlach experiment(1922) - Spin • Heisenberg’s uncertainty principle(1927) • Schrodinger’s wave equation (1926)

u Schrodinger’s wave equation

u Copenhagen interpretation • ψ is a wave of ‘probability’ • Named from the place ‘Copenhagen’ where was a middle of argument • Niels Bohr, Max Born, Heisenberg P(r) P(r) • 1 1 Detecting r r

u Realism • ψ is a wave of quanta itself • Einstein, Schrodinger, De Broglie • Incompleteness of Schrodinger equation - The complete equation will be able to find the exact state of a quanta • “God does not play dice”

u Schrodinger’s cat thought experiment(1935) • Extreme Copenhagen interpretation : Human’s perception affects detecting results • According to the extreme Copenhagen interpretation, there exists an alive ‘and’ dead cat simultaneously → Exclude the human effect in detecting

u EPR(Einstein-Podolsky-Rosen) Paradox(1935) Alice Bob 𝑇 # +1 -1 O X 𝑇 " 𝑇 "

u Bohm’s Hidden variable theory(1952) P(r) P(r) • 1 1 r r P(r) • 1 r

u Hidden variable setting in EPR experiment • Hidden variable λ in a probability space Λ • The values observed by Alice( A ) or Bob( B ) are functions of ⃗ … ∈ 𝑇 ) ) and the λ only the detector settings( 𝑏 ⃗, 𝑐, 𝑑 𝐵, 𝐶 ∶ 𝑇 ) × Λ → {−1, +1} 𝐶(𝑏 ⃗, λ)=−A( 𝑏 ⃗, λ)

u Bell’s inequality • The quantum correlation between A( 𝑏 ⃗, λ) and 𝐶(𝑐, λ), defined as an expectation value of a product of the two components, is C(𝑏 ⃗, 𝑐 ) ≡8 𝑞 λ A ( 𝑏 ⃗, λ) 𝐶(𝑐, λ)dλ= − 8 𝑞( λ ) A ( 𝑏 ⃗, λ) 𝐵(𝑐, λ)dλ ( 𝑞( λ ) : probability density) • If 𝑑 ⃗ is an another detector setting, C(𝑏 ⃗, 𝑐 )− C(𝑏 ⃗, 𝑑 ⃗ )= − 8 𝑞 λ [A ( 𝑏 ⃗, λ) 𝐵(𝑐, λ)− A ( 𝑏 ⃗, λ) A ( 𝑑 ⃗, λ)]dλ

u Bell’s inequality ⃗, λ) 𝐵(𝑐, λ)[1− A ( 𝑑 ⃗, λ) C(𝑏 ⃗,𝑐 )− C(𝑏 ⃗, 𝑑 ⃗ ) = − 8 𝑞 λ A ( 𝑏 ]dλ A ( 𝑐, λ) = 8 𝑞 λ A ( 𝑏 ⃗, λ) 𝐵(𝑐, λ)[ A ( 𝑐, λ) A ( 𝑑 ⃗, λ)−1]dλ |C(𝑏 ⃗, 𝑐 )− C(𝑏 ⃗,𝑑 ⃗ )| ≤ 8 𝑞 λ [1− A ( 𝑐, λ) A ( 𝑑 ⃗, λ)]dλ 1 + 𝐷(𝑐,𝑑 ⃗ ) ≥ |𝐷(𝑏 ⃗, 𝑐 )− 𝐷(𝑏 ⃗, 𝑑 ⃗ )|

u Bell’s inequality simple verification • 8 possible cases of spins electron positron ⃗ ) C(𝑑 ⃗, 𝑏 ⃗ ) C(𝑏 ⃗,𝑐 ) C(𝑐,𝑑 • Calculate C(𝑏 ⃗, 𝑐 ), C(𝑐,𝑑 ⃗ ), a b c a b c -1 -1 -1 C(𝑑 ⃗, 𝑏 ⃗ ) in each case +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 +1 -1 -1 -1 +1 +1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 +1 -1 -1 -1 +1 +1 +1 -1 +1 -1 +1 +1 +1 -1 -1 +1 +1 -1 -1 +1 -1 +1 -1 +1 -1 +1 +1 -1 -1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 +1 +1 +1

u Bell’s inequality simple verification LHS RHS • Calculate LHS and RHS of Bell’s inequality 0 0 2 1 1 + 𝐷(𝑐,𝑑 ⃗ ) ≥ |𝐷(𝑏 ⃗, 𝑐 )− 𝐷(𝑏 ⃗, 𝑑 ⃗ )| 2 1 0 0 • In all cases, the inequality holds 0 0 2 1 2 1 0 0

u Correlation as calculated by Quantum mechanics (𝑏 ⃗, 𝑐 )= < 𝑇 𝜏 ) D 𝑐 𝜏 E D 𝑏 ⃗ 𝑇 > (|𝑇 > = 1 (|χ I > |χ J > −|χ J > χ I > ) 2 (𝜏 " = 0 1 0 , 𝜏 L = 0 −𝑗 , 𝜏 # = 1 0 −1 ) 1 𝑗 0 0 𝐷 N (𝑏 ⃗, 𝑐 ) = − 𝑏 ⃗ D 𝑐 • This doesn’t satisfy Bell’s inequality

u CHSH inequality • John Clauser, Michael Horne, Abner Shimonv, Richard Holt(1969) • Advanced version of Bell’s inequality |𝑇| ≡ |𝐹 P,Q − 𝐹 P,Q R + 𝐹 P R ,Q + 𝐹 P R ,Q R | ≤ 2 (𝐹 P,Q = 𝑂 I,I + 𝑂 J,J − 𝑂 I,J − 𝑂 J,I ) 𝑂 I,I + 𝑂 J,J + 𝑂 I,J + 𝑂 J,I ( 𝑂 I,I : Number of simultaneous occurrences of the outcome +1 on both sides and vice versa • |𝑇| N > 2

u Freedman and Clauser experiment(1972) • First actual Bell test • Using Freedman’s inequality u Aspect et al(1982) • Using photon polarization • 𝑏 ∶ 0°, 𝑏 U ∶ 22.5°, 𝑐 ∶ 45°, 𝑐 U ∶ 67.5°

u Loopholes in Bell test experiment • Detection efficiency / Fair sampling - Inaccurate measurement by coincidental factors ≤ 4 𝐹 P,Q|[\]^[. − 𝐹 P,Q R |[\]^[. + 𝐹 P R ,Q|[\]^[. + 𝐹 P R ,Q R |[\]^[. η − 2 ( η : efficiency of experiment) - If η is less than 83%, there would be no violation with Q.M prediction - Efficiency of typical optical experiments was around 5~30%

u Loopholes in Bell test experiment • Detection efficiency / Fair sampling - Fair sampling assumption : Sample of detected pairs is representative of the pairs emitted → Set η as 1

u Loopholes in Bell test experiment • Locality / Communication - Prohibit any communication by separating the two sites - Measurement duration must be shorter than the time it would take for any light-speed signal from one site to the other, or indeed, to the source

u Hensen et al(2015): “loophole-free” Bell test • Detect two entangled spin of electron which is trapped in nitrogen-vacancy(NV) defect centre in a diamond chip • The diamonds are mounted in closed-cycle cryostats (T=4K) located in laboratories named A and B which distant about 1.3km

u Hensen et al(2015): “loophole-free” Bell test • Constructing entanglement - Event-ready set-up(entanglement swapping) New entanglement 𝑏 𝑐 𝑏 U 𝑐 U 𝑏,𝑏 U 𝑐 U ,𝑐

u Hensen et al(2015): “loophole-free” Bell test • Schematic

u Hensen et al(2015): “loophole-free” Bell test • Space-time analysis of the experiment • Locality - It takes 4.27μs between A and B in speed of light - Measuring duration : 3.7μs < 4.27μs • Detection efficiency - Through 245 trials, result in Figure c - Measuring fidelity A : 97.1 ± 0.2% , B : 96.3 ± 0.3%

u Result • Substitution of experimental values results in violation of CHSH inequality in all experiments • In Hensen’s experiment, 𝑇 = 2.42 ± 0.03 > 2 Hidden Copenhagen variable theory Interpretation

u Violation of special relativity in EPR experiment • Two particles which have an entanglement can interact simultaneously → ‘Non-locality’ quantum characteristic • Many experimental data prove this phenomenon