 Thus, each unit vector of \(\mathbf{H}\) determines a probability weight on \(\mathcal{A}_{\mathbf{H}}\). Quantum mechanics asks us to take this literally: any “maximal” discrete quantum-mechanical observable is modeled by an orthonormal basis, and any pure quantum mechanical state, by a unit vector in exactly this way. Conversely, every orthonormal basis and every unit vector are understood to correspond to such a measurement and such a state.