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

Take the trine state ensembleAll three states are normalized. Since $1+\omega +{\omega }^2=0$, their average density matrix isGeometrically, 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.