Pure-state realization of a basis measurement distribution ($|\psi ⟩={\sum }_j\sqrt{p_j}{e}^{i{\theta }_j}|e_j⟩$)

For an orthonormal basis $\{|e_j⟩\}$ and nonnegative numbers $p_j$ summing to one, the displayed vector is a unit vector and gives measurement probabilities $p_j$ by the Born rule. Conversely, every pure state with those probabilities has coefficients of these magnitudes, with arbitrary phases on the nonzero coefficients. A common change of all phases is a global phase; differences between them are relative phases that can affect measurements in another basis. Zero-probability coefficients vanish, so their phases have no effect.