Schrodinger's dead cat when the evidence is destroyed In the classic (and morbid) Schrodinger's Cat thought experiment, we imagine putting a cat into a box with a vial of poison which will be triggered by a quantum detectors. We set up a radioactive nucleus or an excited ion, and set up a detector to look for radiation resulting from radioactive decay or drop to the ground state. When we detect this radiation, we break a vial of cyanide, which would kill the cat. If the box is sealed away from external influence, it is commonly said that the cat is in a superposition of life or death, as a result of the uncertain time of the radiation occurring: with no-one to observe the cat, the state of its health life itself becomes uncertain in a quantum mechanical way.
Add a twist. Don't ever open the box. Explode it with a bomb instead and destroy all evidence. Did the wave function ever collapse?
 A: You really ought to watch the talk by Charles Bennett at http://pirsa.org/11050052/. At first, he analyzed the Mach-Zehnder interferometer and concluded the which-way information of the photon did not exist. Then, he went on to quantum Darwinism where redundant copies of the information are stored in the environment. This multipartite entanglement with the environment leads to a classical correlation whenever not all the copies are taken into account. Some of the redundant copies of the information escape from the Earth and become inaccessible to us. For the case of your cat, the copies of the information here on Earth eventually thermalize or get carried away by thermal radiation from the Earth, or stored in a relatively permanent form here on Earth in a "hash" encoding which is one-way and inaccessible. Eventually, all the entanglement will only exist outside the Earth and be inaccessible, and by the same logic that led us to conclude the which-way information was in a superposition because we can't access it, the cat was also in a superposition. There's a nice quote by Wheeler in the talk "The past only exists in so far as it is recorded in the present". Not even God can tell what "actually" happened.
All in all, this is a very informative talk, and I think it addresses your question head on.
A: This question strikes close to the heart of The measurement problem, which is the question of what (if anything) the process of measurement represents; and is all but synonymous with the question of how one ought to interpret quantum mechanics.
As such, the answer to this question is (a) subject to debate; and (b) absent any substantial philosophical and/or technical break-throughs, entirely subjective. Nevertheless, one can consider the popular alternatives which have been suggested as conceivable answers to the problem.


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*Marco's earlier answer exemplifies the Bayesian interpretation of quantum mechanics: that the wavefunction does not represent the state of the system, but our knowledge of the state of the system. (I am not sure whether the 'Bayesians' actually form any opinions at all about what the state of the system actually is, if the wavefunction is entirely an object of human conception; I would perhaps naïvely assume that they believe that it bears some pertinent relation to the state of the system.) In this interpretation, the cat presumably is always either alive or dead; a measurement (if one is made) only reveals which is the case.

*Opposite in some respects to the Bayesian interpretation, and identical in other respects, the Bohmian interpretation supposes that the wavefunction is not the state of the system, but is nevertheless a real quantity — a "quantum potential" — which governs by delicate means the actual evolution of the system. For Bohmians as with Bayesians, measurement only reveals the pre-existing state of the system; but the wavefunction does change (smoothly) in the presence of measurement devices (which after all are physical systems).

*The Bayesians and the Bohmians both set themselves quite clearly apart from adherents of the Many Worlds Interpretation of quantum mechanics. Like the Bohmians, Many-Worlders believe the wave-function refers to something real; unlike the Bohmians, they believe it completely specifies the state of the system. Like the Bayesians (at least according to my personal understanding), Many-Worlders suppose that the collapse of the wave-function is only apparent, and due to an updating of the information of an observer's knowledge about the state of a system; unlike Bayesians, however, this updating comes from a physical process entangling the observer with different possible marginal states of the observed system, which prior to observation could not be said to have any definite state (in the sense that "definite state" is usually used).
All three of the above interpretations are ones for which the state of the system is not considered to actually collapse; either the system is in some definite (albeit perhaps unknown) state before measurement, or all measurement outcomes are realized in different alternative worlds.
There are also interpretations of quantum mechanics in which the state of the system is considered to collapse. The oldest of these is the original Copenhagen interpretation, in which (as with the Bayesian interpretation) the wavefunction is considered a useful fiction, but which supposes that there is no state of the system between measurements; the wavefunction is literally nothing more than a book-keeping device for describing the relationships between the results of  a sequence of measurements. Somewhat less anti-realist  is the consistent histories interpretation, which fleshes out the Copenhagen interpretation (and conceivably flirts with the MWI) by saying that that while the state of the system is not well-defined per se, the past states of the system can be described in terms of all those histories which are consistent with past measurement results and is in some sense the sum of them. In both of these interpretations, the wave-function does collapse.
Separate from these are theories which do not interpret quantum mechanics, but seek instead a slight variation of it. Spontaneous collapse theories, for instance, not only describe state-collapse as a genuine way that the system may evolve, but describe it as doing so as a result of things such as significant variations in the wavefunction of mass distributions. Roughly speaking, if the wavefunction leads to a superposition of a needle pointing to one position on a dial versus a substantially different one — or to different distributions of potassium in one location of your visual cortex than another — then the wavefunction undergoes a biased random-walk process in which it eventually becomes stable at a "pointer" state consistent with a definite result of the observation; and the more distinguishable the possible states are in terms of spatial distributions of mass, the faster the walk converges.
So to answer your question: whether the wavefunction collapses under any circumstances at all depends on your interpretation of quantum mechanics, and may in fact not be an answerable question. It ultimately depends on what a measurement actually is, whether it necessarily involves consciousness (pseudoscience alert, ugh!) or is merely a mental process (Bayesians / Bohmians), a process resulting from entanglement of conscious observers with quantum systems (the Many Worlds Interpretation), an entirely physical process of state evoltion (spontaneous collapse theories), or something else.
As for the cat:


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*For the Bohmians and Bayesians, the cat was always either dead or alive (becoming dead at least when the bomb explodes).

*The Many Worlds adherent believes that the cat was for a time in a superposition of dead and alive, becoming only certainly dead once the bomb explodes; and believes that a later observation might entangle an observer with a definite state of the cat having been dead even prior to the explosion, if there were any way to determine any difference between the cat being alive or dead prior to the explosion.

*The spontaneous collapse theorist believes that the different behaviours of the cat, between being alive or dead, would cause the collapse of the state of the cat to definitely alive or definitely dead well prior to the bomb.

*Given that they cannot observe the cat, it is not even clear whether the adherent of the Copenhagen even believes the cat exists once the experiment is engaged, unless an observation is made some time after the bomb explodes. Bohr (the original Copenhagen interpreter) liked to distinguish between "quantum" systems and "classical" ones; surely a cat (or a particle detector) is a 'classical' system, and so perhaps they think that the cat is always either alive or dead, but because the particle detector collapsed the wavefunction of the radioactive nucleus / excited ion.

*Consistent histories, like the Many Worlds Interpretation, does not posit any definite state of the system until it is measured, at which point it is revealed (if any distinguishing evidence can be found) that it was either definitely alive before the bomb, or definitely dead. If no distinguishing evidence can be found, no definite status of life before the bomb can be fixed.


Doubtless I have left out an interpretation or two from this list; but in any case, the short answer is "we don't know". 
A: I have to go with the comment of Peter Shor here. There is a collapse, but it's not to the dead/alive-prior-to-bombing basis, but to the basis defined by the configuration of radioactive dust.
This immediately suggests a modification of this thought experiment making it delayed choice. Wait a long time after the cat is poisoned or not, without bombing or opening the box and taking a peek. Then, the experimenter makes an arbitrary choice, perhaps randomly, as to whether to open the box or bomb it. If he opens the box, the collapse is to the dead/alive preferred basis, or as I like to call it, preferred projectors because it is really into an orthonormal set of eigenspaces, not vectors. And I have to add, decoherence requires coarse graining, and so, it will always be preferred projectors. If he chooses to bomb it, the collapse is to some radioactive dust projectors. Here comes the interesting part. Both sets of projectors don't commute. This is a death blow strike to the many worlds interpretation. If MWI is right, which projectors did the cat split into before the experimenter made the choice? The choice of preferred projectors is always contextual.
If you are interested, read more about unhappening by the physicists Susskind and Bousso.
A: The wave function represents our knowledge of the state of the system not the state of the system itself.
What every schoolboy knows about Schrodinger's cat is often inaccurate and obtained from popular literature; poorly explained and misleading.
A: Quantum Darwinsim can answer your question. The focus is turned from the system, here the cat or the box to the environment. It is not that a copy of the information is stored in the environment. Rather, multiple redundant copies are stored in many fragments of the environment, each of which have become decoupled from each other and the system.
You want something better than a mere box. Boxes are not soundproof and multiple copies of the air vibrations of the live cat walking about and meowing and purrrrrrrrring will spread all over to the environment in all directions. Anyone can sample just a little of the soundwaves, some fragment, and learn the cat is alive. No, you have to put the cat in a deep underground bunker. Although come to think about it, if no one is in the room where the box is in, and there are no video cameras or tape recorders, the sound waves would reflect off the walls a couple of times, then get absorbed, and all the fragmented copies rethermalize. If a tree falls in a forest and no one hears it, did it make a sound?  You also want the bomb to be so good that not even the best forensic experts can figure out what went on. You need a nuclear bomb.
Quantum evolution is unitary. The information about the cat always survives in some form, but highly scrambled. The question is if information is stored in easily extractible form in multiple copies in small fragments. Multiple copies for error correction and easy access by multiple independent observers. Intersubjective agreement as the basis for objectivity. If the information is so scrambled that you need to sample a significant fraction of the environment followed by fine tuned computations to extract the information, that is not classical. It is not decohered.
Think of a black hole. Suppose you throw the box into a black hole. The evidence will leak out in the form of Hawking radiation, but Don Page had shown you need to collect half of the Hawking environmental radiation before you can even extract it in principle, but not in practice. You will not find multiple copies stored in multiple small fragments. There can be no decoherence. It is not about the cat. It is about what multiple fragments of the environment knows about the cat.
A: What a metaphysical question. Are there any empirically measurable differences between the following three cases? 
the cat never collapsed 
the cat collapsed into aliveness
the cat collapsed into death
By stipulation, no evidence whatsoever remains after the bomb went off. In practical terms, this means all three scenarios give exactly the same empirical predictions. Any interference between the live and dead cat states will be so suppressed by decoherent phase cancellations as to be undetectable. For All Practical Purposes, FAPP, your question is totally irrelevant. Do you really think you can have access to the wave function and its values and you can go around and poke at it to find out the answer? Now go out and do something practical and make a practical calculation.
A: It's true, decoherence will prevent us from interfering the cat with itself. But that's totally missing the point. We're not even making any pretense of interfering the cat. We're blowing up the cat, for goodness sake!
Get this straight! In the "quasiclassical" basis, off-diagonal terms are exponentially suppressed. But get this too: There are also tons of other bases where off-diagonal entries are also highly suppressed. There is nothing special about the "quasiclassical" basis in this respect. Read your Dowker and Kent.
The issue isn't whether or not interference terms are highly suppressed. They are. The issue is there are way way too many bases where this is the case. There's nothing special about "quasiclassical" bases, you know. Under the "democratic" "egalitarian" principle, any set of consistency histories satisfying the consistency conditions is "equally" valid. Picking out the "quasiclassical" basis in advance to analyze the system is like looking up the answers to a homework problem in advance, and only then writing down a solution contrived to get that answer. That's considered CHEATING!
A: There is a variant of the moveable Heisenberg cut, except that the cut lies in time, and not space or scale. Basically, you might think the collapse happened at a certain time, like during the detection by the Geiger counter, or when the cat's brain registers the cyanide, whatever, but you can always push the cut farther and farther forward in time. It shouldn't make any difference. You can always assume the collapse in the far future "retroactively" fixes the outcome at earlier times via a "delayed choice" mechanism if you wish. How far forward in time can you push the cut? Right until the Ultimate Observer's Mind and/or the end of time.
And in fact, if you are scrupulous, you have the obligation to push the cut as far forward in time as you can as there is always a tiny nonzero probability (exponentially suppressed is still nonzero) that recoherence might happen in the future.
In the OP's example, information about the cat leaks outward from the box in fragments which aren't picked up by any human or recording device or forensic analyst. Eventually, all of these information fragments leave Earth and travel outward in space until they cross the cosmological horizon, never to be accessible by the Ultimate Observer. So, if the Ultimate Observer's Mind knows nothing about the cat, it doesn't exist. 
According to Copenhagenists, nothing becomes real until it reaches the Mind of the One and Only Ultimate Observer. Unless the poor mortal cat happens to be the One and Only Ultimate Observer, the cat didn't exist after it was sealed in the box. Out of sight, out of mind!
A: It helps to look at your question from the point of view of Consistent Histories. Some time back, Dowker and Kent made a startling discovery couched in mathematical terms. However, in a more handwavy manner, we are not allowed to ask about the specifics of dinosaurs roaming 65 million years ago based upon what little fossil records we have remaining.
A brief overview of Consistent Histories first; Von Neumann postulated that the wave function collapes according to some set of orthonormal projectors adding up to the identity operator. Consistent Histories tries to extend this to a history of such sets of orthonormal projectors at different times, which do not necessarily commute with each other. So, we take the time ordered product of operators to get histories.
$C_{i_1 \cdots i_N} \equiv P_{i_1,1} \cdots P_{i_N,N}$
Consistent Histories ought to be weakly consistent so that for different histories, $i \neq j$, $Tr[ C_i \rho C_j^\dagger ] \approx 0$. Basically, off-diagonal entries are exponentially suppressed compared to diagonal entries.
Let $C_{i_2,2}$ be some current present day coarse-grained history available to us, including fossil records and modern day descendents of dinosaur species, plus whatever else you might also want to add. Let $C_{i_1,1}$ be some specific dinosaur history, like whether a dinosaur roamed at some particular area at some specific day 65 million years ago. Of course, if you make the history sufficiently coarse-grained to the mere existence or nonexistence of dinosaurs 65 million years ago, barring some massive conspiracy, we get some weakly consistent histories. On the other hand, what we are interested in is some further refinement of it. Let $C_{j_1,1}$ be an alternative dinosaur history. Precisely because very few records of dinosaur history are available from our present accessible coarse-grained history, we come to the conclusion that $Tr [ C_{i_2,2}C_{i_1,1} \rho C_{j_1,1}^\dagger C_{i_2,2}^\dagger ]$ is not weakly consistent. In some hypothetical alternate universe where dinosaurs were intelligent and developed a writing system and some very advanced technology to store their writing in time capsules which can last reliably for 65 million years, and some dinosaur journalist jotted down his observations in the time capsule, then weak consistency becomes possible. Alas, dinosaurs were very dumb.
How can this be? When the founders of Consistent History, Hartle and Gell-Mann were confronted with this, they claimed they knew that all along, but they never stressed that. This is their dirty little secret of Consistent Histories, a skeleton in their closet.
Their resolution was a copout. They introduced the notion of incompatible realms. There is the present day realm which only includes present day history and no specific dinosaur history, and another dinosaur realm with only dinosaur history but no human history. Each realm in itself is weakly consistent but there can be no refinement including both realms. They are dumping the objective reality of dinosaur history down the drain.
There is also another realm combining quasiclassical present histories with crazily mixed projection bases for dinosaurs scrambled like a spaghetti, mashed up far worse than a superposition of live and dead cats. No one in their right might will take it seriously.
In general, the past does not commute with the present. Neither does the present commute with the future. The exception is when detailed records from an earlier time survive intact until the latter time. The past belongs to a distinct realm from the present.
What does this have to do with Schrödinger's cat with a bomb? Destroying the evidence makes the cat's fate the same as the dinosaurs. Incompatible realms. A cat realm including the quasiclassical coarse-grained history of the cat before the bomb exploded but not any quasiclassical coarse-grained history after that, and an outsider realm including the quasiclassical history of observers after the bomb explosion but nothing about the quasiclassical history of the cat between the time the killer contraption was set and the time the bomb went off.
Incompatible realms. That is the skeleton in the closet. Which history does the wave function collapse into? Choose your realm first.
