Notes on Preskill chapter 4

Joe Gittings

June 2001

4.1.1 Hidden quantum information

Bell states:

| ${\Phi }^1$ ⟩ = $\frac{1}{\surd 2}$ (|00⟩1|11⟩ )

| ${\psi }^1$ ⟩ = $\frac{1}{\surd 2}$ (|01⟩1|10⟩)

4.1.2 Einstein locality and hidden variables

Hidden-variable theory result for measuring the pure state| ${\uparrow }_z$ ⟩along axis rotated byθfrom z:

If we assume | ${\uparrow }_z$ ⟩ is in fact parameterized by (z,λ) where 0 ≤ λ ≤ 1 is the hidden variable, the outcome is:

| ${\uparrow }_{\theta }$ ⟩ for 0 ≤ λ ≤ ${\cos }^2\frac{\theta }{2}$

| ${\downarrow }_{\theta }$ ⟩ for cos $2^{}\frac{\theta }{2}<\lambda \le 1$

If λ is unknown, the probability distribution for the measurement agrees with quantum theory.

4.1.5 More Bell inequalities

Most general statement of Bell inequality:

For two photons whose polarizations are correlated in the state | ${\Phi }^{+}$ ⟩:

|⟨ab⟩ - ⟨ac⟩| ≤ 1 - ⟨bc⟩ is violated

where the observables are

a = ${\tau }^{\left(A\right)}$ (α)

b = ${\tau }^{\left(B\right)}$ (β)

c= ${\tau }^{\left(A\right)}$ (γ) = ${\tau }^{\left(B\right)}$ (γ)

and τ(θ) is the polarization operator with eigenvalues 11

i.e. a is the polarization of photon A measured along the axis α.

CHSH (Clauser-Horne-Shimony-Holt) inequality:

|⟨ab⟩ +⟨a' b⟩ + ⟨a' b'⟩ - ⟨ab'⟩| ≤ 2

where a, a', b, b' = 11

is violated by quantum theory

4.1.6 Maximal violation

Cirelson's inequality:

||C|| ≤ 2√2

where

C = ab + a' b + a' b' - a b'