 Quantum state ray

A quantum state ray is the equivalence class of nonzero vectors in a complex Hilbert space under multiplication by a nonzero complex number. Its span is a one-dimensional linear subspace, but the zero vector is not a state representative. A unit vector fixes the norm and leaves a global phase free. ^[ref]

For a nonzero representative $|v⟩$, the associated pure state density operator isReplacing $|v⟩$ by $\lambda |v⟩$ multiplies both numerator and denominator by $|\lambda {|}^2$, so the quotient is unchanged. In particular, for every POVM effect $E$,This proves that a global phase changes no measurement probability. Changing the relative quantum phase of two nonzero components can change the density operator, as the following comparison shows.