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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

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

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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