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

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.

• <rant> Dead is not a state vector, and neither is Alive. The common statement that the cat is in a superposition of dead and alive states is misleading.</rant> It might in principle be possible to detect a superposition of macroscopic objects, but I agree that it's hard to see how this could actually be done. Feb 28, 2015 at 8:18
• Why the downvote? Aside from my rant about dead/alive superposition this seems a perfectly reasonable question. Feb 28, 2015 at 11:18
• Your rant is probably exactly why it got a downvote (although it wasn't from me). Schroedingers Cat is so misrepresented in popular culture and media, we have a bunch of people walking around thinking that it is, in practice, possible to prepare dead and alive cat
– Sean
Feb 28, 2015 at 12:47
• A problem with the question is how do you define dead/alive? Isn't this quite difficult even in classical context? Feb 28, 2015 at 12:52
• OP, please clarify if you are asking: "Are there measurement techniques to detect whether a single state is a superposition?" or "Are there measurement techniques to detect whether a macroscopic object is in a superposition?" or "Are there techniques to detect whether Schrödingers cat is in a superposition of dead and alive?". As John Rennie says, the latter question is nonsense, but the first two would be equally fine. Feb 28, 2015 at 14:05

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.
• Or, even better, define $U$ so that $U\k{dead}=+\frac1{\sqrt2}\left(\k{alive}-\k{dead}\right)$, then you'd get $U^2\k{alive}=\k{alive}$, i.e. 100% revival :) . Jul 2, 2015 at 9:44