Two notions of a pure state Let $\mathscr{H}$ be a complex infinite-dimensional separable Hilbert space and let $\mathscr{S}$ be the state space associated to $\mathscr{H}$, which is defined as the set of all positive (hence self-adjoint) trace class operators $M$ such that $\operatorname{Tr}M = 1$. In addition, pure states are defined as projection operators from $\mathscr{H}$ to one-dimensional subspaces of $\mathscr{H}$.
The above objects arise in mathematical quantum mechanics. However, in the physics literature, the Hilbert space $\mathscr{H}$ itself is called the state of pure states.
Question: Is there a connection (isomorphism, maybe?) between $\mathscr{H}$ and the space of pure states (as defined above) that justifies using the same name for both objects?
In addition: why is the trace a nice way to define quantum states? Intuitively, I'd think of them as being elements of the underlying Hilbert space; in the physics literature, a state is often given by a wavefunction for instance.
 A: As user953376 pointed out in their comment one-dimensional subspaces are determined by a unit vector. To fill in some detail: as an application of the Riesz representation theorem for every bounded linear operator $T$ on $\mathscr H$ the image of which is a one-dimensional, there exist $\psi,\phi\in\mathscr H$ such that $T=|\phi\rangle\langle\psi|$ (that is, $T(z)=\langle\psi,z\rangle\phi$ for all $z\in\mathscr H$). Now if $T$ is a self-adjoint projection then $T=|\psi\rangle\langle\psi|$ and $\langle\psi,\psi\rangle=1$. Thus every pure state in the above sense is determined by a unit vector, giving rise to the mapping
\begin{align*}
\iota:\{\psi\in\mathscr H:\langle\psi,\psi\rangle=1\}&\to\{T\in\mathcal B(\mathscr H):T\text{ orthogonal projection},\operatorname{dim}(\operatorname{ran}(T))=1\}\\
\psi&\mapsto|\psi\rangle\langle\psi|\,.
\end{align*}
While $\Psi$ is surjective, it fails to be injective (again for the reason user953376 described in their comment): $\psi$ and $e^{i\alpha}\psi$ are mapped to the same operator under $\iota$. This is solved by replacing the domain of $\iota$ by equivalence classes $[\psi]$ under the equivalence relation $\psi\sim\psi':\Leftrightarrow \psi=\psi' e^{i\alpha}$ for some $\alpha\in\mathbb R$---then $\iota$ is the isomorphism you were looking for.
As for your second question: there are two main reasons to consider states to be positive semi-definite trace-class operators of unit trace---or, equivalently, positive normal functionals $f$ on a $W^*$-algebra such that $f({\bf1})=1$---as opposed to mere elements in a Hilbert space:

*

*As seen before, defining states via vectors would mean that global phases make a difference (i.e. $\psi\neq e^{i\alpha}\psi$ for any $\alpha\in(0,2\pi)$). But such phases can never be detected experimentally because they vanish in the expectation value $\langle\psi|A|\psi\rangle$ and thus in the measurement.

*Secondly and more importantly, looking at trace class operators allows for a description of mixed states. Such states are convex combinations of pure states and usually arise whenever relaxation or other environmental effects enter the picture. For example the maximally mixed state for a qubit $\operatorname{diag}(1/2,1/2)=\frac12|0\rangle\langle 0|+\frac12|1\rangle\langle 1|$ describes that there is a 50% chance of the state becoming the first eigenstate ($|0\rangle\langle 0|$) after measuring it and, similarly, a 50% chance for the system to be in the second eigenstate ($|1\rangle\langle 1|$) after measurement. Now a description of states via elements of $\mathscr H$ falls short of such features for the eigenvalue (probability) of $|\psi\rangle\langle\psi|$ is always equal to one, so there is "nothing else" in $\mathscr H$ which could describe mixed states and obey a quantum-mechanical probabilistic interpretation.

To expand on this second point: every state $M$ (positive semi-definite trace class operator of unit trace) on an arbitrary Hilbert space allows for a spectral decomposition (because trace class) $M=\sum_{j=1}^\infty p_j|\psi_j\rangle\langle\psi_j|$ with "probabilities" $p_j\in[0,1]$ (positivity), $\sum_{j=1}^\infty p_i=1$ (trace condition) and an orthonormal system $\{\psi_j:j\in\mathbb N\}$ in $\mathscr H$. Therefore the pure states are fundamental to the concept of states as those are the "building blocks", but in order to get the full picture one has to allow for mixtures of such states, which are mathematically described by such convex combinations.
