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

u Before 20th century

• Classical Mechanics
• Absolute space & time
• Matter = particle
• Light = wave

Isaac Newton Thomas Young Particle Wave Young’s double-slit experiment

u 20th 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) r

1

P(r) r

1

• Detecting

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

O

𝑇" u EPR(Einstein-Podolsky-Rosen) Paradox(1935)

Alice Bob

𝑇#

+1

• 1

𝑇"

X

u Bohm’s Hidden variable theory(1952)

P(r) r

1

P(r) r

1

• P(r)

r

1

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

the detector settings(𝑏 ⃗, 𝑐, 𝑑 ⃗… ∈ 𝑇)) and the λ only 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 C(𝑏 ⃗,𝑐)−C(𝑏 ⃗, 𝑑 ⃗) = − 8 𝑞 λ A(𝑏 ⃗, λ)𝐵(𝑐,λ)[1− A(𝑑 ⃗, λ) A(𝑐, λ) ]dλ = 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

electron positron a b c a b c +1 +1 +1

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+1 +1

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+1 +1

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+1

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+1

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+1

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+1 +1

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+1 +1 +1

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+1

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+1

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+1

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+1 +1 +1

• 1
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+1 +1 +1

• 8 possible cases of spins

C(𝑏 ⃗,𝑐) C(𝑐,𝑑 ⃗) C(𝑑 ⃗, 𝑏 ⃗)

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• 1
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+1 +1 +1 +1

• 1

+1

• 1

+1 +1

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+1 +1 +1

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+1 +1

• 1
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• 1
• Calculate C(𝑏

⃗, 𝑐), C(𝑐,𝑑 ⃗), C(𝑑 ⃗, 𝑏 ⃗) in each case

u Bell’s inequality simple verification

• Calculate LHS and RHS of Bell’s inequality

1 + 𝐷(𝑐,𝑑 ⃗) ≥ |𝐷(𝑏 ⃗, 𝑐)−𝐷(𝑏 ⃗, 𝑑 ⃗)|

• In all cases, the inequality holds

LHS RHS 2 1 2 1 2 1 2 1

u Correlation as calculated by Quantum mechanics (𝑏 ⃗, 𝑐)= < 𝑇 𝜏) D 𝑐 𝜏E D 𝑏 ⃗ 𝑇 > (|𝑇 > = 1 2 (|χI > |χJ > −|χJ > χI > ) (𝜏" = 0 1 1 0 , 𝜏L = 0 −𝑗 𝑗 , 𝜏# = 1 −1 ) 𝐷N(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,QR + 𝐹 PR,Q + 𝐹 PR,QR | ≤ 2 (𝐹 P,Q = 𝑂I,I + 𝑂J,J − 𝑂I,J − 𝑂J,I 𝑂I,I + 𝑂J,J + 𝑂I,J + 𝑂J,I ) (𝑂I,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°, 𝑏U ∶ 22.5°, 𝑐 ∶ 45°, 𝑐U ∶ 67.5°

u Loopholes in Bell test experiment

• Detection efficiency / Fair sampling
• Inaccurate measurement by coincidental factors

𝐹 P,Q|[\]^[. − 𝐹 P,QR|[\]^[. + 𝐹 PR,Q|[\]^[. + 𝐹 PR,QR|[\]^[. ≤ 4 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)

𝑏,𝑏U New entanglement 𝑏 𝑐 𝑐U,𝑐 𝑏U 𝑐U

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

Copenhagen Interpretation Hidden variable theory

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