quantum mechanics - What's the influence due to the phase factor in state vector?

In the fundamental postulate of quantum mechanics (page 18), there exists a one-to-one correspondence like that:

$\text{closed quantum system $\leftrightarrow$ Hilbert space ; quantum state $\leftrightarrow$ a ray in Hilbert space }$

but here the ray can carry some additional phase factor $e^{i\phi}$? What's the observable influence due to the phase factor in state vector? I mean if you take inner product the redundant factor will disappear.