How to check whether Schrödinger's cat was in superposition of states?

Question

Suppose we can make an arbitrarily precise preparation of a Schrödinger's cat (and isolate it arbitrarily well so that decoherence is not a problem). If we prepare lots of cats in this state, what measurement can tell us whether these cats were in a superposition of dead/alive states or just in a mixture of them? Of course, I mean experiment not on the poison-triggering decaying particle, but on the cat itself.

I guess we need some sort of double-slit or other experiment which would let us see interference between dead/alive states, but I can't seem to come up with an experiment suitable for such a large "particle" as a cat.

To clarify: let's model our "cat" as an object with two possible states, like a supercooled fluid and a crystal. So the supercooled fluid would correspond to the cat which is alive, and crystal would be a dead cat. I.e. in this model the decaying particle would create a nucleation site.

The question then is how to experimentally distinguish the state $\alpha\left|\mathrm{liquid}\right\rangle+\beta\left|\mathrm{crystal}\right\rangle$ from the state $\alpha\left|\mathrm{liquid}\right\rangle\left\langle\mathrm{liquid}\right|+\beta\left|\mathrm{crystal}\right\rangle\left\langle\mathrm{crystal}\right|,$ given that we can prepare arbitrary number of copies of the system in that state.

Answer 1

The problem is that

suppose we can make an arbitrarily precise preparation of a Schrödinger's cat

is not quite what you need to do. You need to extend your capabilities so that you can make an arbitrarily precise implementation of the unitary which takes $\newcommand{\k}[1]{\left|\,\mathrm{#1}\right\rangle}\k{alive}$ to $U\k{alive}=\tfrac1{\sqrt{2}}(\k{alive}+\k{dead})$. If you want to be able to detect this, you need to stay reasonably confined to a small dimensionality, which then implies by orthogonality that $$U\k{dead}=-\tfrac1{\sqrt{2}}(\k{alive}-\k{dead})$$

The idea is then to run your cat twice through the black box, which will invariably give $$U^2\k{alive}=\k{dead}.$$ This is impossible to achieve with purely incoherent population driving: if your box kills live cats 50% of the time and revives dead cats 50% of the time, then a 50-50 dead/alive mixture can only result in a 50-50 dead/alive mixture. In essence, you're running Rabi oscillations instead of incoherent decay and driving.

This method is going to be pretty hard to implement with an actual cat, insofar as our current technology prevents us from raising the dead. With a more reversible transition, like your liquid-crystal one, there's more hope - you just need to be careful and coherent in your control of both directions of the transition.