Every definition of the two is always very abstract to me. Like, A pure state is located on the surface of the bloch sphere while the mixed state is somewhere within.
First of all, what is an intuitively definition?
And second of all, how do you practically recognize whether a given state is mixed or pure?
A pure state is a ket in Hilbert space. A mixed state is a probabilistic mixture, which cannot be described by a single ket. The importance of the concept is illustrated in Bell tests, which show that quantum mechanics (i.e. the result of calculation using Hilbert space) cannot be replicated by classical probability theory. In practice, I do not know of any other examples giving a clear empirical distinction. I don't find it helpful to worry about it too much. The concept of mixed states was due to von Neumann, who was more concerned with matter of principle than practicality.
A quantum state is pure if you know as much as one can know about a quantum state.
A quantum state is mixed if you could know more about, that is, you don't know as much as is allowed to know about a quantum state by the laws of quantum mechanics.
