# Unitarily reversing a projective measurement

We start with a particle in a pure superposition state. Let's say it is,

$$\vert\psi\rangle = \frac{1}{2}(\vert 0\rangle + \vert 1\rangle)$$

Alice sends this particle inside a box and the box performs a projective measurement in the standard basis. Say, the outcome of the measurement is $$\vert 0 \rangle$$. Note that this outcome gives us classical information and can be cloned. This classical information (possibly after cloning) is sent from the box to Alice.

Can we now reverse the measurement process where the system "box + environment" can be made to go back into the pre-measurement state $$\vert\psi\rangle$$ but the classical information "0" is retained by Alice?

Some clarifications

1) The state of Alice, the box and the environment after the projective measurement is $$\vert 0\rangle_A\otimes\sigma_{BE}$$. I would like to go from this state to $$\vert 0\rangle_A\otimes\vert\psi\rangle_{BE}$$, where $$\vert\psi\rangle$$ was the state before measurement occurred.

2) The state $$\vert\psi\rangle$$ is unknown in general. Otherwise, the box can trivially generate copies of the state since it knows a description of it.

The motivation for asking about the reversing the process is because I am thinking of the measurement process in the box as a unitary process on a bigger space (the box + environment). If one has access to the environment, can one in principle reverse the measurement but still keep the outcome "0" that was obtained?

• What precisely do you mean by "reverse process"? What should the box do? Is the state |psi> known? etc. ... – Norbert Schuch Oct 4 at 21:37
• @NorbertSchuch, thanks for the questions. I have edited the question to clarify :) – user1936752 Oct 5 at 12:20

If you could go back the way you describe it, nothing would prevent Alice from measuring $$|\psi\rangle$$ again, again, and again, in whatever basis she likes. At the same time, as per your protocol she would keep full record of the past measurement outcomes.
This would therefore allow her to perform full tomography of $$|\psi\rangle$$, and thus obtain a full description of it, using only a single copy, which is impossible. Therefore, such an operation cannot exist.