# can entangled particles return to superposition?

Let's suppose we have two entangled particles that can either have the values of 1 or 0 when measured. Before measurement, both particles are in a superposition of both 1 and 0 and have a 50/50 chance of being 1 or 0. When one particle is then measured, then the other particle will have the opposite measurement of the first. So if one is 0 then the other will measure 1. But after measurement, if we now stop measuring both particles, can they both go back to the superposition state of being 0 or 1 and still remain entangled?

• A measurement brings the pair to an eigenstate of whatever measurement you made. From there, the state evolves according to the Schrodinger equation. If the Hamiltonian is "unentangled" (i.e. of the form $H\otimes J$ where $H$ and $J$ act on the state spaces for the individual particles), then it can never bring an unentangled state back to an entangled one. (Past history is of course irrelevant.) – WillO Dec 6 '18 at 19:08
• I think there is a really nice variant of this question for a research context, phrasing it in terms of weak measurements and the resulting decoherence... but probably the answer is still "no"? – CR Drost Dec 6 '18 at 21:33

## 2 Answers

Immediately after the measurement they are disentangled, as you say. (The technical term is that their joint state is then a product state.) In order to entangle them again, you would need a physical apparatus which involved either the two particles directly interacting, or else some influence that moved from one of them to the other. This can be done (in practice it involves some experimental ingenuity). It is significant to note that the speed of the re-entangling process is limited by the speed of light.

After measurement, a state in a superposition "collapses" to one of the states with some probability. After this collapse, nothing stops you from trying to turn that new collapsed state back into an entangled state - but the origional state has been "lost".

This is related to the no cloning theorem, and other ideas. Basically if we could revert a collapsed state back to its origional uncollapsed state (without knowing what our uncollapsed state is), then we would be able to uncover the wavefunction using only a single uncollapsed state (something that is impossible by various theorems. Basically, you always multiple identical unknown states to recover their underlying probability function, in the same way that you need to flip a coin more than once to know if it's a "fair coin").

• +1 for the emphasis on the irreversibility of the information loss, most presentations in the subject in the current age barely gloss over that fact and leave students the impression that "conservation of information" actually holds in our physical universe – lurscher Dec 6 '18 at 21:28
• "nothing stops you from trying to turn that new collapsed state back into an entangled state" - does this assume that the particles have to be in close proximity or can you re-establish this entangled state at a distance. – kishdude Dec 6 '18 at 22:22
• This is answered by @Andrew Steane - They need to be in contact or have some kind of contact in a chain. For instance, if you want to keep particle A and B separated, you can move particle C to A, entangle them, then move particle C (physically) near B, then entangle these states. By measuring/manipulating the state of C, you can manipulate the entanglement of C and A. – Steven Sagona Dec 6 '18 at 22:45
• This is known as "quantum teleportation" or entanglement swapping – Steven Sagona Dec 6 '18 at 22:45