Main Question or Discussion Point
Are all quantum states represented by normal vectors?
Major slip of the brain there, I should have read my post properly. What I meant to ask is: Are all quantum states represented by unit vectors in some complex vector space?Every quantum state is represented by a vector in an abstract vector space.
What exactly do you mean by "normal"?
Is this what is referred to as a global phase factor?No, pure states are never represented by unit vectors but by unit rays (i.e. by unit vectors modulo an arbitrary phase factor)
So is the direction of a vector component the defining feature for a pure state? This makes sense because two rays with the same direction but different magnitude would give the same expectation value with respect to the same observable since they are identical unit vectors once normalized. Right?I would say that they are represented by rays, where a ray is a set of all vectors that are a complex constant multiple of one another (same "direction", different magnitudes).
I thought that pure states corresponded to points on the bloch sphere which are defined by unit vectors aren't they?It is an important point to make clear that pure states are represented as rays not as vectors in Hilbert space.
They appear because in reality states are positive operators of unit trace - not elements of a vector space. By definition pure operators are of the form |u><u|. The |u> are elements of a vector space and can, and usually are, thought of that way. But note |cu><cu| = |u><u| where c is any complex number of unit length ie a phase factor. Thus phase factors are unimportant.So what exactly are they there for or why do they appear?
Because a pure state is a ray, when you write a unit vector representing the state, there is more than one way to write it, and all states that differ by a global phase factor are different ways of writing the same state. It is analogous to the electric potential in electrostatics - only the potential difference is physical - physical situations described by potentials that differ by a global constant represent the same physical situation. Since the global phase factor of the wave function and the global constant to the potential can be freely chosen according to taste, convention and convenience.I must admit I've had a difficult time getting to grips with these global phase factors. Im currently writing a project on the basics of quantum information and quantum computing and they crop up everywhere. I understand that global phase factors are essentially irrelevant since they do not affect the expectation values for any observables. My proffesor has brushed over the toic when I hav mentioned it. So what exactly are they there for or why do they appear?