Is the collapse of the wave function inherently time asymmetric? Schroedinger's equation, as we all know, is time symmetric. In quantum field theory, we have to come up with a more sophisticated CPT reversal, but the essential point remains unchanged. However, the collapse of the wave function in the Copenhagen interpretation is manifestly time asymmetric. Correct me if I'm wrong, but can you uncollapse a wave function, converting it from an eigenfunction to a superposition of eigenfunctions?
Is this asymmetry connected with the thermodynamic arrow of time and the second law in statistical mechanics, or are they independent? How would an uncollapse look like, and can we experimentally arrange for an uncollapse? Why are there more collapses than uncollapses? If an observer unobserves a quantum thingie, does that thingey uncollapse?
 A: The collapse of the wavefunction is generally attributed to decoherence. This is time asymmetric in the same way the second law of thermodynamics is time asymmetric. I suppose it's theoretically possible for a wavefunction to uncollapse, but this is like saying it's theoretically possible for a broken egg to reassemble itself.
A: As suggested in the answer above, in general, decoherence increases the entropy associated with a quantum system and as such has the same type of time-reversal asymmetry that appears in thermodynamics. The question, however, is also concerned with how an "uncollapse" would look like. Here I want to illustrate how this can be done in principle.
The net effect of a projective measurement on a pure quantum system is a nonlinear mapping from an initial state $|\psi\rangle$ to a final state $|\psi'\rangle$. The nonlinearity arises from the fact that the final state must be normalized.
Nevertheless, what is important is that the final state is also a pure state of unit norm, and there always exists a reversible unitary mapping connecting the two. Hence, it is possible to simply apply the reverse unitary on the state after the measurement to get back the original state.
Here is an example. Suppose we start with the state $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and measure in the computational basis $\{|0\rangle,|1\rangle\}$. After the measurement, the state will be described either by $|0\rangle$ or $|1\rangle$ with equal probability. For the sake of argument let's assume it is $|0\rangle$. 
Then all we need to retrieve our original state is to apply a Hadamard transform 
$H=\left(\begin{array}{cc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}  \\\\
\frac{1}{\sqrt{2}} &  -\frac{1}{\sqrt{2}}
\end{array}\right)$
to retrieve the initial state. This follows from the relation
$H |0\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$. Note that something like this can be routinely implemented in the lab.
In the case of mixed states and general measurements the situation is a bit more complicated, but by introducing an auxiliary system one could also perform a mapping between the state after a measurement and before it.
A: I had the same thought a few years before you did. As I understand it - and I'm still thinking about this one - your question is excellent - when this happens out and about, all information about what the system was is lost within the system itself when it 'collapses' due to a measurement. There is nothing in the system itself which retains information about its history. Hence the 'random' choice of one of the eigenstates and values. 
Suppose there is the case of a system which just absorbs a photon - this will change the wavefunction by shifting its energy up a photon's worth, but it is now in a new eigenfunction of energy. This also therefore changes its momentum and thus its position is differently unknown. But how do we know of these unknowns? We can only find out by the measurement of the system - so how do we know about its new energy eigenstate? We'd have to make another measurement. Suppose we did so by chucking another photon in its direction and conveniently it chucks one back out in our direction which we absorb (and we assume nothing else is interacting with the system). Now our energy eigenfunction has shifted - perhaps we can tell this by being the measuring apparatus (say we are a frog with a good eye for one photon). 
So the two systems have interacted through that single-photon interaction. And because the frog's eye saw a blue photon, we know it was that energy that came out of the system rather than the energy that a red photon would indicate. So we know the system just dropped from whatever state it had evolved to (a mess of its energy eigenstates) at its previous energy eigenlevel down one blue photon eigenlevel. If we knew already (from a previous measurement) what that eigenlevel was, we know what it just dropped to, and also by The Laws of Quantum Physics, that it's once again supposedly now time-evolved to another mess of its eigenstates. But we can't know that something else didn't do something else with it in-between, like, say, the exact same thing we were doing to it but from a different direction, and perhaps in the absorption and emission of a red photon instead - and you see the difficulties about knowing what's going on.
And that is the simplest version of how all such interactions must take place. But we didn't get any information about what the system had evolved to before that blue photon came to us, and none is retained in the system. Some would say that information never really existed at all (because it's only information upon interaction), and therefore the system was never in any state other than an eigenstate by reasoning of what information can be gained about a system. 
I would then ask about the effectiveness of quantum computation - and then be told it's multiple universes, in each eigenlevel-numerated of which the system is in the appropriate eigenstate, but this still begs the question of how the universes know how to communicate such that they don't accidentally send the information about one 'collapse' to the wrong universe, or if you like, what controls the propagation of universes, and how it is that we can get the result we're looking for out in this universe by dragging back the interference from all the others. Actually, the loss of information in this universe is clearer to see in this model, because the other eigenstates are so well-removed in the other eigenlevel universes (as well as in their other times, if that's still bothering you).
In the case of the laboratory described in a previous example, we know what is going on there because we have set up that experiment and know its properties. We can therefore 'reverse' what happened to it because what we are doing in that case, is exactly the same as starting the experiment again. Note it isn't the same experiment - there is no such thing in quantum physics - it's done at a different time.
I have perhaps been a little verbose to detail my description. But I think such discussion require such attention. I reiterate that your question is excellent, and that I, too, wonder about this time directionality problem and whether it is all controlled at precisely this level. Controlled by what is, I think, the next big step in physics.
A: The simplest way to answer this question is to rephrase it in the Heisenberg Picture where states have no time-dependency.
The two von Neumann axioms for quantum theory are only ever posed in the Schrödinger Picture: Evolution (quantum states evolve in time in accordance with the Schrödinger Equation) and Projection (each measurement of a physical value results in a quantum event that projects the state to one of the eigenstates of that value, in accordance with the Born Rule).
The Measurement Problem is that of explaining what the Born Rule actually is and does; when, where and how. The Interpretations of Quantum Theory are the different answers to that question.
Various alternate interpretations have arisen (Decoherence, Many Worlds, Consistent Histories, etc.) whose purpose is to entirely "explain away" the Born Rule. In them, of necessity, collapse is reversible (although not necessarily macroscopically reversible!) Other interpretations, objective collapse formalisms, treat it as an actual dynamic process separate from that which is described by the Evolution postulate. Penrose and Diosi are notable members of the latter camp. In them, collapse is probably not reversible (depends on the dynamics). What they all have in common is that they provide a way to hybridize quantum and classical systems and dynamics so as to overcome a no go result which prohibits back-reaction from the quantum to the classical sectors. The collapse is registered in the classical sector as a process that is macroscopically irreversible and probably even microscopically irreversible.
"Quantum-classical hybrid dynamics: a summary"
https://iopscience.iop.org/article/10.1088/1742-6596/442/1/012007/pdf
"A no-go theorem for theories that decohere to quantum mechanics"
https://arxiv.org/abs/1701.07449
"Statistical consistency of quantum-classical hybrids"
https://arxiv.org/pdf/1201.4237.pdf
"A post-quantum theory of classical gravity?"
https://arxiv.org/abs/1811.03116
There was one reply here that used the passive voice weasel word in "The collapse of the wavefunction is generally attributed to decoherence." (emphasis mine) to lend false authority to one of many answers of this question, which is not settled. (https://en.wikipedia.org/wiki/Weasel_word) Always be watchful of that ploy, because that's the vehicle by which the meme virus of dogmatism is spread.
Actually, Decoherence was never intended by its originators as a complete explaining away of the Born Rule, only something that gets you 90% of the way and leaves the final 10% of the problem for posterity to resolve. So, it's something you inject into other interpretations, but provides no interpretation for or resolution of the measurement problem in its own right ... and certainly no final answer to the question of irreversibility!
"Decoherence and its role in the modern measurement problem"
https://royalsocietypublishing.org/doi/full/10.1098/rsta.2011.0490
"Why Decoherence has not Solved the Measurement Problem" https://arxiv.org/abs/quant-ph/0112095 (And yes, it persuaded Anderson, who it was addressed to, that he was wrong.)
So, as I was saying, all you need to do is ask what everything looks like in the Heisenberg Picture.
In the Heisenberg Picture: if collapse is reversible, then the state (which, in the Heisenberg Picture, is time-independent) will remain the same before and after the collapse. If the collapse is irreversible, then the state will jump to a different state. In the latter case, in effect, all time-dependency of states (and probably even all perception of time as a flow) would reside in the second von Neumann postulate, rather than in the first!
So, to completely resolve the issue (on an interpretation-by-interpretation basis), all you need to do is work out the Born Rule or any of the interpretations meant to partially or completely replace it (e.g. Decoherence, Consistent Histories, Many Worlds, Many Minds, etc.) or dynamicize it (e.g. Penrose/Diosi) look like in the Heisenberg Picture.
So, let's do a look up (e.g. ArXiv, Google Scholar) and see. Hmm, that's odd. There's one minor problem here.
It looks like someone was asleep at the switch here and forgot to answer the question: what do {Born Rule, Decoherence, Consistent Histories, Many Worlds Interpretation, Objective Collapse} look like in the Heisenberg Picture?
What does physical wave function collapse theories, like Penrose/Diosi, look like in the Heisenberg Picture? I can't seem to find too much here either. Open systems dynamics, Lueders Rule might be places to look.
Bear in mind, the two pictures are only "equivalent" on the one issue: the Evolution postulate (it's the Schrödinger Equation in the Schrödinger Picture and the Heisenberg equations of motion in the Heisenberg Picture). They're not equivalent on the other issue, the Projection postulate, because ... there is no Born Rule in the Heisenberg Picture at all! It was never formulated in that picture: it's still an open problem. To express it in the Heisenberg Picture, you need extra infrastructure: (1) a point cloud in space-time consisting of all the points where a quantum event, witnessing an application of the Born Rule, takes place, (2) a network of Heisenberg states that each partition the point cloud into "before" and "after" subsets (with none of the "after" points residing in the causal past of any of the "before" points), (3) a postulate that asserts that any two states that agree on their before/after sets in all but 1 point, will have between them a transition given by a single-instance Born Rule that takes place at that 1 point.
Interpretations, that try to explain away the Born Rule or dynamicize it, will have to somehow account for this infrastructure, before they can address the issue of whether each of the transitions in (3) is reversible or not.
So, we can't really answer the question, until people finish writing down the Heisenberg Picture versions of each of these interpretations.
A: The non-reversibility of the wave function collapse, if theoretically true, will establish a clear distinction between time as a mere set of real numbers that can sequentially track a set of events in a motion or a process and time as we consciously experience it, i.e as past, present and future wherein the past can never be revisited and the present constantly vanishes into the past, never to come back again.
Even if we reversed the rotation of the earth or of its orbit around the sun, the reversal is merely a reversal of motion or proces and would make absolutely no difference to the way we consciously clock the progress of time. The direction of such natural motions upon which our theoretical time is baed is actually completely arbitrary.
That is why I believe that iff the wave function collapse is irreversible it can be used as a good model our conscious sense of time i.e. the non- reversible kind.
A: Depends on the interpretation. Under Everett/manyworlds/decoherence theory the Universal Wave Function evolves unitarily/deterministically and is entirely reversible. No different than standard thermodynamics, where the illusion of irreversibility is merely statistical due to the evolution from low to high entropy.  However with objective collapse theories such as Copenhagen, the evolution over time is irreversible under measurement (the UWF superposition collapses) and information is lost.
