 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2012/paper-1/2/a/solution

An ensemble of normalized quantum states with probabilities

p_{j}

is represented by the density operator

ρ = j \sum p_{j} ∣ ψ_{j} ⟩ ⟨ ψ_{j} ∣, p_{j} \geq 0, j \sum p_{j} = 1.

It determines every observable expectation value through

⟨ A ⟩ = Tr (ρ A)

. The ensemble decomposition need not be unique; the operator specifies the measurement statistics.

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