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[itex]\mid \Psi \rangle=\frac{1}{\sqrt{2}}(\mid alive \rangle+\mid dead \rangle)[/itex]

This is representing a particular pure state which is distinctly different from

[itex]\mid \Psi' \rangle=\frac{1}{\sqrt{2}}(\mid alive \rangle-\mid dead \rangle)[/itex]

In some sense, these are different rotations of the alive-dead spinor.

But, should this be more properly treated as a density matrix?

[itex]\begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix}[/itex]

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?