Schrödinger's cat, pure or mixed state?

[Khashishi]

The usual description of Schrödinger's cat is that after being placed in the box, Schrödinger's cat in a superposition state of being alive and dead. Sometimes, I see this written as:
$\mid \Psi \rangle=\frac{1}{\sqrt{2}}(\mid alive \rangle+\mid dead \rangle)$
This is representing a particular pure state which is distinctly different from
$\mid \Psi' \rangle=\frac{1}{\sqrt{2}}(\mid alive \rangle-\mid dead \rangle)$
In some sense, these are different rotations of the alive-dead spinor.
But, should this be more properly treated as a density matrix?
$\begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix}$
That is, we have incomplete information about the Schrödinger's cat's exact state, which is significantly less mysterious than claiming the cat is alive+dead.

We could, for example, construct a Schrödinger box by loading an ampoule with poison, and triggering the poison with a photon detector which is behind a polarizer and a photon source. The photon source emits randomly polarized light. The photon polarization is in a mixed state, so the cat should be, too. Can we actually construct a pure state of cat actually be in a pure state of alive+dead?

 

[Demystifier]

You are right, Schrodinger cat is in the mixed state. The photon is also in the mixed state. But the cat, photon, photon-detector, and their environment TOGETHER are in a pure state.