Schrodinger's Cat Bloch Sphere

A diagrammatic view of the wavefunction of the ubiquitous Schrödinger's cat.
[link]

The wavefunction can be any complex-valued superposition of alive/dead. Each of alive and dead have a coefficient corresponding to how alive or how dead the cat is, giving 2 degrees of freedom. A third degree of freedom (taking us to 3D) is that choosing to add or subtract alive from dead gives us a relative phase between them, hence one might presume that the states of the cat would span all of 3-dimensional space. However, we require that the wavefunction is normalised such that whatever state it is in, the probability is 1, hence the vector in 3-space must have a magnitude of 1, giving us a sphere: the Bloch Sphere.

EDIT: Upon recent thinking (23/04/2013), I have decided that the cat is NEVER IN A SUPERPOSITION state at all, thus can't have it's state shown on a Bloch sphere. The reason is that when you get to the half-life of the isotope and believe it is now 50-50 that the cat is alive or dead, this is NOT the same as 1/sqrt(2)*(|Alive>+|Dead>)! Since this is only one of an infinite number of states which could give the same 50-50 probabilities (any state on the xy-plane of the sphere gives this, such as 1/sqrt(2)*(|Alive>-|Dead>), see), so which do we choose? The answer is we can't choose any based on our information, because the cat's state wasn't *coherently* transformed from |Alive> towards |Dead>, it is simply a random process. Hence, the cat is not in a superposition state, but in a mixed state and hence there is NO PARADOX! The 50-50 is just a measure of our certainty of whether the cat is alive or dead, not implying that the cat is half-alive and half-dead.

FURTHER: What say we replaced the radioactive decay with a *coherent* process, then? Then the paradox will be renewed, right? Wrong. If we have it as a coherent process (like say a rotating electron spin in an external magnetic field) then how do we have it trigger the poison? We could say that the poison triggers when the electron spin is up, due to an RTS signal on an charge-detector that senses the electron's spin-dependent tunnelling if and only if it is 'up' [link]. However, this 'triggering' process is, by necessity, a projective measurement and hence the coherence is destroyed, falling back to just a probabilistic experiment based on our knowledge of 'how much' we flipped the electron and hence how probable it is that it tunnelled. Either way, whether it's an isotope or an electron spin, it ends up being no more 'quantum' than having a man in a sealed room roll a dice and decide to shoot the cat if it comes up even.
...except perhaps that we'd hear the gunshot, better if he use a silencer.

Comments and questions welcome.