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

Fororthonormality of the $|b_i⟩$ givesNormalization follows from ${\sum }_i{\lambda }_i=1$, so this is a purification of a density operator.

With $|\Omega {⟩}_{RA}=n^{-1/2}{\sum }_i|i{⟩}_R|i{⟩}_A$,The partial trace over $R$ isIts norm is $\mathrm{Tr}{\rho }_A=1$. The positive square root of an operator $\sqrt{{\rho }_A}$ is unique, proving the claim.

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