I see a lot written about pure and mixed states regarding state vectors and density matrices/operators that contain a finite number of states/elements.
For something like the continuous state vector of position $|\psi\rangle = \int_{-\infty}^{\infty} \psi(x) |x\rangle dx$, where $\psi(x)$ is the position wave function and $\int_{a}^{b} \rho(x) dx=\int_{a}^{b} \psi^*(x)\psi(x) dx$ yields the probability of measuring the object within $[a,b]$. I think that we could also speak of a continuous density matrix/operator $\hat\rho(x) = |\psi\rangle \langle\psi|$ such that $\rho = \langle x|\hat\rho|x \rangle$ and when $\hat\rho$ is inserted into the resolution of the identity integrated over all x the result is $1$, confirming the $100\%$ probability that the object is somewhere in space.
For continuous position, is it meaningful to speak of pure vs. mixed states? I would think that a mixed state is where we have $\rho(x)$ but not $\psi(x)$? That might happen after entanglement? Consider the double slit experiment, using photons, where we put different polarizing filters over each slit and then have a radial $\psi(x)$ emerging from each slit, but not adding together. In that case, we would have $$ \rho(x) = \frac{1}{2}(\psi^*_1(x)\psi_1(x) + \psi^*_2(x)\psi_2(x)). $$ Is it not possible to find some $\psi_3(x)$ such that $$ \frac{1}{2}(\psi^*_1(x)\psi_1(x) + \psi^*_2(x)\psi_2(x)) = \psi^*_3(x)\psi_3(x)? $$ I believe that for QM, we demand that any $\psi(x)$ be a "test function", such as a Gaussian or a smooth function of compact support. Because of the restriction on $\psi(x)$, maybe there are situations where there is no such $\psi_3(x)$...