Bell's Theorem
This disproves Einstein, Podolsky, and Rosen (1935)
Let Alice have two possible measurements A 1 A_1 A 1 A 2 A_2 A 2 B 1 B_1 B 1 B 2 B_2 B 2
Each measurements returns ± 1 \pm 1 ± 1
A i , B j ∈ { ± 1 } A_i,B_j\in\{\pm 1\} A i , B j ∈ { ± 1 } Local realism assumes that before every measurement, every observable has a pre-existing Definite value determined by some hidden variable λ \lambda λ A 1 , A 2 , B 1 , B 2 A_1,A_2,B_1,B_2 A 1 , A 2 , B 1 , B 2
Let Q Q Q
Q = A 1 ( B 1 − B 2 ) + A 2 ( B 1 + B 2 ) Q=A_1(B_1-B_2)+A_2(B_1+B_2) Q = A 1 ( B 1 − B 2 ) + A 2 ( B 1 + B 2 ) Given that the measurement of A i , B j A_i,B_j A i , B j ± 1 \pm 1 ± 1 ± 2 \pm 2 ± 2
Therefore, Q Q Q
− 2 ≤ E [ Q ] ≤ + 2 -2\leq \mathbb{E}[Q]\leq +2 − 2 ≤ E [ Q ] ≤ + 2 expand Q Q Q
E [ Q ] = E [ A 1 B 1 ] − E [ A 1 B 2 ] + E [ A 2 B 1 ] + E [ A 2 B 2 ] \mathbb{E}[Q] = \mathbb{E}[A_1 B_1] - \mathbb{E}[A_1 B_2] + \mathbb{E}[A_2 B_1] + \mathbb{E}[A_2 B_2] E [ Q ] = E [ A 1 B 1 ] − E [ A 1 B 2 ] + E [ A 2 B 1 ] + E [ A 2 B 2 ] ∣ E [ A 1 B 1 ] − E [ A 1 B 2 ] + E [ A 2 B 1 ] + E [ A 2 B 2 ] ∣ ≤ 2 \big|\mathbb{E}[A_1 B_1] - \mathbb{E}[A_1 B_2] + \mathbb{E}[A_2 B_1] + \mathbb{E}[A_2 B_2]\big| \leq 2 E [ A 1 B 1 ] − E [ A 1 B 2 ] + E [ A 2 B 1 ] + E [ A 2 B 2 ] ≤ 2 This is CHSH inequality Clauser, Horne, Shimony, and Holt (1969) .
Consider a rotated observable
Let
W θ ( A ) = σ A X sin θ + σ A Z cos θ W^{(A)}_\theta = \sigma_A^X \sin\theta + \sigma_A^Z\cos\theta W θ ( A ) = σ A X sin θ + σ A Z cos θ W θ ′ ( B ) = σ B X sin θ ′ + σ B Z cos θ ′ W^{(B)}_{\theta'} = \sigma_B^X \sin{\theta'} + \sigma_B^Z\cos{\theta'} W θ ′ ( B ) = σ B X sin θ ′ + σ B Z cos θ ′ Combining gives
E [ W θ ( A ) W θ ′ ( B ) ] = E [ ( σ A X sin θ + σ A Z cos θ ) ( σ B X sin θ ′ + σ B Z cos θ ′ ) ] \mathbb{E}[W^{(A)}_\theta W^{(B)}_{\theta'}] = \mathbb{E}[(\sigma^X_A\sin\theta + \sigma^Z_A\cos\theta)(\sigma^X_B\sin\theta' + \sigma^Z_B\cos\theta')] E [ W θ ( A ) W θ ′ ( B ) ] = E [( σ A X sin θ + σ A Z cos θ ) ( σ B X sin θ ′ + σ B Z cos θ ′ )] = sin θ sin θ ′ E [ σ A X σ B X ] + cos θ cos θ ′ E [ σ A Z σ B Z ] + sin θ cos θ ′ E [ σ A X σ B Z ] + cos θ sin θ ′ E [ σ A Z σ B X ] = \sin\theta\sin\theta'\,\mathbb{E}[\sigma^X_A\sigma^X_B] + \cos\theta\cos\theta'\,\mathbb{E}[\sigma^Z_A\sigma^Z_B] + \sin\theta\cos\theta'\,\mathbb{E}[\sigma^X_A\sigma^Z_B] + \cos\theta\sin\theta'\,\mathbb{E}[\sigma^Z_A\sigma^X_B] = sin θ sin θ ′ E [ σ A X σ B X ] + cos θ cos θ ′ E [ σ A Z σ B Z ] + sin θ cos θ ′ E [ σ A X σ B Z ] + cos θ sin θ ′ E [ σ A Z σ B X ] See Singlet
= − sin θ sin θ ′ − cos θ cos θ ′ = -\sin\theta\sin\theta' - \cos\theta\cos\theta' = − sin θ sin θ ′ − cos θ cos θ ′ = − cos ( θ − θ ′ ) = -\cos(\theta - \theta') = − cos ( θ − θ ′ ) Suppose Alice and Bob chose
A 1 = W 0 , A 2 = W π / 2 , B 1 = W π / 4 , B 2 = W 3 π / 4 A_1 = W_0, \quad A_2 = W_{\pi/2}, \quad B_1 = W_{\pi/4}, \quad B_2 = W_{3\pi/4} A 1 = W 0 , A 2 = W π /2 , B 1 = W π /4 , B 2 = W 3 π /4 Where θ = 0 \theta=0 θ = 0 σ Z \sigma^Z σ Z θ = π / 2 \theta=\pi/2 θ = π /2 σ X \sigma^X σ X
E [ A 1 B 1 ] = − 1 2 , E [ A 1 B 2 ] = + 1 2 , E [ A 2 B 1 ] = − 1 2 , E [ A 2 B 2 ] = − 1 2 \mathbb{E}[A_1 B_1] = -\tfrac{1}{\sqrt{2}}, \quad \mathbb{E}[A_1 B_2] = +\tfrac{1}{\sqrt{2}}, \quad \mathbb{E}[A_2 B_1] = -\tfrac{1}{\sqrt{2}}, \quad \mathbb{E}[A_2 B_2] = -\tfrac{1}{\sqrt{2}} E [ A 1 B 1 ] = − 2 1 , E [ A 1 B 2 ] = + 2 1 , E [ A 2 B 1 ] = − 2 1 , E [ A 2 B 2 ] = − 2 1 Substitute into Clauser, Horne, Shimony, and Holt (1969) to get
= ∣ − 1 2 − 1 2 − 1 2 − 1 2 ∣ = 2 2 = \big|{-\tfrac{1}{\sqrt{2}}} - \tfrac{1}{\sqrt{2}} - \tfrac{1}{\sqrt{2}} - \tfrac{1}{\sqrt{2}}\big| = 2\sqrt{2} = − 2 1 − 2 1 − 2 1 − 2 1 = 2 2 However, this contradicts as
2 2 ≰ 2 2\sqrt{2} \not\leq 2 2 2 ≤ 2 The shows that the two are mathematically incompatible . Bell's theorem in concrete form.
2 2 2\sqrt{2} 2 2 Clauser, Horne, Shimony, and Holt (1969) value a state can get.
See how it is exploited with Greenberger, Horne, and Zeilinger (1989)
Clauser, Horne, Shimony, and Holt (1969)