Quantum state population ($p_k=⟨k|\rho |k⟩$)

For a chosen normalized level state $|k⟩$, its population is the probability $⟨k|\rho |k⟩=\mathrm{Tr}(|k⟩⟨k|\rho )$ of finding the system in that state. In an orthonormal level basis these diagonal entries are nonnegative and sum to one. Off-diagonal quantum coherence contains information absent from the population vector, so matching all level populations need not prepare a specified coherent superposition.