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Motivation

Usually the measurement is denoted by for example:

$$ \alpha |0\rangle+\beta |1\rangle \to |1\rangle $$

Note that there is no mention of when this measurement happened. I was wondering if one could peg a number to the "when?"

Revisiting Wigner's Friend

Assume for simplicity that the physical system is a two-state spin system $S$ with states $|0\rangle _{S}$ and $|1\rangle _{S}$ , corresponding to measurement results $0$ and $1$.

Initially, $S$ is in a superposition state

$${\displaystyle \alpha |0\rangle _{S}+\beta |1\rangle _{S}}$$

and gets measured by Wigner's friend $(F)$ in the $\{ |0 \rangle_S, |1\rangle_S \}$ basis. Then, with probability $|\alpha|^2$, $F$ will measure $0$ with probability $|\beta|^2$ will measure $1$.

From the friend's point of view, the spin has collapsed into one of its basis states upon his measurement, and hence, they will assign to the spin the state corresponding to their measurement result: If they got 0, they will assign the state $|0 \rangle $ to the spin, if they got $1$, they will assign the state $|1 \rangle_S$ to the spin.

Wigner $(W)$ now models the combined system of the spin together with his friend (the joint system is given by the tensor product $S\otimes F$) . He thereby takes a viewpoint outside of $F$'s laboratory, which is considered isolated from the environment. Hence, by the laws of quantum mechanics for isolated systems, the state of the whole laboratory evolves unitarily in time. Therefore, the correct description of the state of the joint system as seen from outside is the superposition state

$$ \alpha (|0\rangle _{S}\otimes |0\rangle _{F})+\beta (|1\rangle _{S}\otimes |1\rangle _{F}) $$

where $|0\rangle_F$ denotes the state of the friend when they have measured $0$, and $|1 \rangle_F$ denotes the state of the friend when they have measured $1$.

For an initial state $|0\rangle_S$ of the state for $S \otimes F $ would be $|0\rangle _{S}\otimes |0\rangle _{F}$ after $F$' s measurement, and for an initial state $| 1 \rangle_S$, the state of $S \otimes F$ would be $|1 \rangle_S \otimes |1 \rangle_F $. * Now, by the linearity of Schrödinger's quantum mechanical equations of motion, an initial state $\alpha |0\rangle _{S}+\beta |1\rangle _{S}$ for $S$ results in the superposition $\alpha (|0\rangle _{S}\otimes |0\rangle _{F})+\beta (|1\rangle _{S}\otimes |1\rangle _{F})$ for $S \otimes F$.

So the usual Wigner's Friend is spoken of in the context pure states. But let's say the state in question was a state described by a density matrix $\hat \rho_s$ represented in some basis $\hat \rho_s(|\psi_s \rangle)$.

Using the same argument in the wiki link the density matrix of the joint system would be as modelled (?) by Wigner (W):

$$\hat \rho_W = \hat \rho_s(|\psi_s \rangle \otimes |\psi_F \rangle) $$

Note: using the arguments in the link the friend should also come to this conclusion.

Question

My question is if Wigner can determine at what time the measurement his friend took place? Since the time evolution is governed by the unitary operator as highlighted in line *

(?) I think this is reasonable?

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