Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 2 / a / ii / 3 / Solution

The states ${\rho }_A^i=|a_i⟩⟨a_i|$ are pure, so a purification isTracing out the orthonormal reference labels removes all cross terms and recovers the stated classical-quantum mixture. Its rank is the number of nonzero $p_{ij}$, so that is the minimum reference dimension for a particular state. The smallest dimension that can purify every state of the stated form is