Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 2 / c

Table of contents
- Solution c

Solution

X, Z

are real symmetric,

tr (X^{2}) = tr (Z^{2}) = 2

, and

tr (XZ) = 0

. Part b therefore gives

⟨ Ω∣ A \otimes B ∣Ω ⟩ = cos α cos β + sin α sin β = cos (α - β) .

Since the local measurement outcomes are

\pm 1

, their product is one exactly on agreement. Thus

E [ab] = 2 P (a = b) - 1

, and

P (a = b) = \frac{1 + cos ( α - β )}{2} = cos^{2} (\frac{α - β}{2}) .

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