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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?

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I would love to hear someone address the connection to the arrow of time head-on in their answer. – kleingordon May 13 '12 at 1:25
I think in Road to Reality Roger Penrose concludes the collapse of the wave function is time asymmetric – Anant Saxena Jul 19 at 21:09

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.

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well of course decoherence is the answer, but how could it be theoretically possible for a broken egg to reassemble itself? It is not a cartoon running backwards! – anna v May 11 '12 at 13:03
If you have a system in a certain a state $\psi_n$, you can make a superposition by affecting it with a time dependent external force (a push). "Collapse" of a superposition is not only in "picking up" one of the states of the superposition, but often also a destruction of the state, like absorption of a photon. – Vladimir Kalitvianski May 11 '12 at 14:29
anna v: i think conservation of phase space volume or in other words conservation of information guaranties that theoretically it is possible for broken egg to go back to its initial state. In fact the probability of such thing happening is not zero! it is small but not zero! , – user55867 Aug 24 '15 at 6:11
Or to put it in other words: direction of time is not violation of time riversal symmetry, look at this post by Sean Caroll in his blog Time-Reversal Violation Is Not the "Arrow of Time", – user55867 Aug 24 '15 at 6:21
@John Rennie ... It may be that the wikipedia article has been edited since then but it clearly says: "Decoherence does not generate actual wave function collapse. It only provides an explanation for the observation of wave function collapse, as the quantum nature of the system "leaks" into the environment." – Anant Saxena Jul 19 at 21:08

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.

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Everything here sounds correct, but the mention of the second law of thermodynamics in the OP's question makes me suspect he/she is concerned with systems with many degrees of freedom, which become much harder to time-reverse. – kleingordon May 13 '12 at 1:27
I agree that the question might be concerned with a general scenario in which we allow more complex systems that are harder to time-reverse. My intention was only to explicitly show how this can be done in principle and provide an example for a simple case. I guess I should have been more clear about the goal of my answer and I have changed this! – Juan Miguel Arrazola May 14 '12 at 18:17
Thanks, and +1. Good answer – kleingordon May 14 '12 at 20:59

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.

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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.

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