# Space of states in quantum mechanics

A state in quantum mechanics I think is just a vector in a complex Hilbert space. As the physical properties are defined up to a phase $e^{i\theta}$ then this Hilbert space is invariant under the action of $S^1$ and the quotient which what is physically relevant is just an infinite complex projective space. Is this right?

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A more modern view (see, e.g. Bengtsson and Zyczkowski, 2008): the quantum state space is the convex set of trace 1 positive semidefinite operators. That is $\rho \in L(\mathcal H)$ (space of linear operators acting on $\mathcal H$) which satisfy $$\rho\geq 0, \text{Tr}(\rho) = 1.$$ The extreme points of this convex set are the projectors onto 1-dimensional subspaces. These pure states can be characterized as the states additionally satisfying $$\text{Tr}(\rho^2) = 1.$$ These can be put in one-to-one correspondence with the elements of the complex projective Hilbert space $P(\mathcal H)$.