 Solution

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

The given real symmetric matrix is Hermitian and has trace one. A spectral decomposition supplies its positivity without relying on diagonal entries alone:

ρ = \frac{3}{4} ∣ v_{1} ⟩ ⟨ v_{1} ∣ + \frac{1}{4} ∣ v_{2} ⟩ ⟨ v_{2} ∣, v_{1} = \frac{( 2 , 1 , 1 ) ^{T}}{6}, v_{2} = \frac{( 0 , 1 , - 1 ) ^{T}}{2} .

The remaining normalized eigenvector is

(- 1, 1, 1)^{T} / 3

with eigenvalue zero. Consequently the spectrum is ${3/4, 1/4, 0}$ , every eigenvalue is nonnegative, and the matrix is permissible. Its purity is

(3/4)^{2} + (1/4)^{2} = 5/8

, so it is mixed.

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