Can a quantum measurement be made undone?

Question

Can a quantum measurement be made undone, so that the original superposition reappears? Let me illustrate my question with an example.

Consider the Schrödingers cat thought experiment, in which the cat in the box is in a superposition $\frac12\sqrt2(|\mathrm{dead}\rangle + |\mathrm{alive}\rangle)$. When I open the box, the system collapses into one of the two states; let's say it is $|\mathrm{alive}\rangle$. Now suppose that I completely forget the result of the experiment, remove all data about the cat from my lab, and close the box. If I succeeded in reversing the measurement, the state of the cat would again be $\frac12\sqrt2(|\mathrm{dead}\rangle + |\mathrm{alive}\rangle)$. However, I am not sure if this is the possible, because the cat state did collapse at some point. Has the cat been reduced to a 'classical' uncertainty, where the cat is dead or alive with probability 0.5? Or can I really invert the measurement and recreate the superposition?

Answer 1

No. Quantum measurements are irreversible. See:

When is a quantum measurement? by Asher Peres, American Journal of Physics 54, 688 (1986); doi: 10.1119/1.14505

Quoting from the abstract:

The measurement of a quantum system is not consummated until irreversible processes have destroyed all phase coherence between different possible outcomes of that measurement.

In the simplest cases, this is mathematically expressed by the fact that projection operators (which describe measurements) do not have inverses.

Answer 2

There is a subtlety here that depends on your interpretation of what "collapse" actually means.

In the many-worlds interpretation, when you observe the state of the cat you become entangled with it, so your combined wavefunction becomes $\frac{1}{\sqrt 2}(|dead\rangle|obs. dead\rangle + |alive\rangle|obs. alive\rangle$. This interaction is completely described by the unitary time evolution (i.e. Schrödinger's equation), so in principle it can be reversed back to the original wavefunction.

Note however that this implies that you own state will have to be reversed as well, so from your subjective point of view, you will not "see" the cat's state return to a superposition state. Rather, your own state will return to the state in which it was before you made the measurement.

Answer 3

Returning to the quantum superposition is impossible to perform in practice, as the atoms in your body will be entangled with the result through decoherence. You cannot remove this effect. Therefore, forgetting the result will make it a 'classical' uncertainty.

Answer 4

In general no, but in some cases part of the quantum state can be returned to the original state.

Consider e.g. spin 1/2 state of a silver atom, which initially is known to be $|y+\rangle$. Let the spin projection on $x$-axis be measured on this atom by an SG magnet; this measurement prepares the state randomly into state $|x+\rangle$ or state $|x-\rangle$, and sends the atoms into one of two directions accordingly. Downstream in both directions, there can be a device that can produce magnetic pulse of the right direction, intensity and length, that it rotates the spin state back into $|y+\rangle$ state.

This recreates the spin state, but not the complete quantum state, because the spin state is recovered only after the atom is downstream from the apparatus, and moving in one of two different directions.

Answer 5

In classical physics the equations of motion of measurable properties of a system such as the $x$ position are written in terms of a function $x(t)$ such that if you measure $x$ at time $t$ you will get the value $x(t)$.

In quantum physics the equations of motion of a system are written in terms of matrices called observables whose eigenvalues are the possible results of measuring that quantity. Quantum theory predicts what the probability of each of those possible values would be if you measured it.

In general the outcome of an experiment depends on what happened to all of the possible values of the measured quantity. This is called quantum interference. For an example see Section 2 of

https://arxiv.org/abs/math/9911150

A measurement is an interaction that creates a record of some property of the measured system. Such interactions can and do take place without any human intervention, e.g. - light reflecting from a cat when you open a box, changes in the cat's position exerting pressure on the walls of a box etc. These interactions suppress interference: a process called decoherence:

https://arxiv.org/abs/1911.06282

In practice these most of these interactions can't be reversed for any object you can see in everyday life and you would have to undo all of them to undo decoherence. If you measure a single photon onto a single atom or something like that, those systems are relatively isolated and controllable and the interaction might be reversible to a good approximation, see

https://arxiv.org/abs/1808.08598

Answer 6

Here is the easiest way to see that what you're suggesting is impossible:

Write $D$ for the dead state and $A$ for the alive state. Put two identical cats in identical boxes. Put one of them in state $D+A$ and the other in state $D-A$.

You open both boxes, examine both cats, and by chance you find them both alive. They are now both in the state $A$. There is now no difference between these cats.

Put them back in their boxes. Because there is no difference between them, their states have to evolve identically. If the first cat evolves to state $D+A$ then the second cat evolves to state $D+A$ --- which is not at all the same thing as state $D-A$. So at least one of your cats must fail to evolve back to its original state.

(Repeat if you like with arbitrarily many cats all in different states of the form $D+e^{i\theta} A$ for real values of $\theta$ that are distinct mod $2\pi$. Then of your vast number of cats, only at most one can evolve back to its original state.)