I am wondering, what is the dimension of a ray in a Hilbert space? For example here (page 2, bottom of page) I have read: I understand why a state is represented by all multiples of a vector, not just the vector. But is the ray really one-dimensional? It would be one-dimensional if we multiply the vector by real numbers. But we are multiplying it by complex numbers. Along one-dimensional ray we could fully determine the vector multiple by a single real number. However with complex numbers there is one additional degree of freedom. Thus for each position along one-dimensional ray there are infinitely many choices for one additional parameter. Or "one-dimension" here means a complex dimension, i.e. 2 real dimensions?