I have some confusion (uncertainty? :)) about the compatibility between states and superposition in quantum mechanics. I give a bit of background context and then ask my question at the end of the 4th paragraph.
In the mathematical description of quantum mechanics, here is what I understand states of a system to be. First, they may be described by a unit vector (wave function) $\psi$ in a Hilbert space, with the proviso that all unit vectors of the form $e^{i\theta}\psi$ describe the same state as $\psi$. Second, we could say states of the quantum system are described by nonzero vectors $\psi$ not necessarily of unit length, with the proviso that all of its scalar multiples $c\psi$ for nonzero $c \in \mathbf C$ describe the same state as $\psi$. (The standard convention to focus on states as unit vectors is made to simplify the probabilistic interpretation given by the Born rule, I suppose.) Third, since nonzero vectors that are not scalar multiples of each other are supposed to describe different states of the system, the states can be described without any redundancy as the $1$-dimensional subspaces of the Hilbert space.
The states of the quantum system can undergo superposition: we are allowed to form linear combinations of states and this leads to new states (with dynamics described by Schroedinger's equation). This is why vector spaces are the framework in which quantum states are described.
However, different $1$-dimensional subspaces of a vector space can't be added to get a single $1$-dimensional subspace as a result: if $\psi_1$ and $\psi_2$ are two linearly independent vectors in the Hilbert space, then $\psi_2$ and $-\psi_2$ are supposed to represent the same state but $\psi_1 + \psi_2$ and $\psi_1 - \psi_2$ (more generally, $\psi_1+\psi_2$ and $\psi_1 + c\psi_2$ where $c \not= 1$) are not scalar multiples of each other so they don't belong to the same $1$-dimensional subspace. The set of all linear combinations $\{a\psi_1 + b\psi_2\}$ is a $2$-dimensional subspace, not a $1$-dimensional subspace. Therefore it seems that superposition is not something that can be done to states, as it would not be well-defined. Yet all accounts of QM that I have looked at refer to "superposition of quantum states". The individual $\psi$'s in the Hilbert space are not supposed to be physically meaningful on their own, while the $1$-dimensional subspaces they each span (the rays of the Hilbert space) are physically meaningful, yet the superposition concept makes sense as an operation on the $\psi$'s but makes no sense on the $1$-dimensional subspaces. How is this conundrum resolved? That is my question here.
In contrast to my confusion over states as $1$-dimensional subspaces not being compatible with superposition, there is "no problem" working with $1$-dimensional subspaces when we form a composite of two quantum systems, since the mathematical description of composite systems uses tensor products and a tensor product of $1$-dimensional subspaces is very naturally a $1$-dimensional subspace again.