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 Incoming links: Trine state ensemble

Past exam of the computer science course of the University of Oxford / 2021 / Quantum information / 3 / b / ii / Solution 2026-10-25

Take the trine state ensemble

∣ t_{j} ⟩ = \frac{∣0 ⟩ + ω ^{j} ∣1 ⟩}{2}, p_{j} = \frac{1}{3}, j = 0, 1, 2, ω = e^{2 πi /3} .

All three states are normalized. Since

1 + ω + ω^{2} = 0

, their average density matrix is

\frac{1}{3} \sum_{j = 0}^{2} ∣ t_{j} ⟩ ⟨ t_{j} ∣ = \frac{1}{6} \sum_{j = 0}^{2} (1 ω^{j} ω^{- j} 1) = \frac{I}{2} .

Geometrically, their unit Bloch vectors form an equilateral triangle on the equator and sum to zero. Orthogonality is unnecessary for an ensemble decomposition of a mixed state: three distinct, nonorthogonal pure states can give the same maximally mixed average as two orthogonal states.

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