I am curious how one can use a density matrix in the following familiar cases in QM.
1. EPR
Suppose a typical EPR situation where the state of a two-electron system is given by
$\alpha|1\rangle|2\rangle+\beta|2\rangle|1\rangle$.
How do you get the state of the $individual$ electron using a density matrix?
2. Another Situation
Suppose a different situation that there is an electron of which state is prepared to be $\alpha|1\rangle +\beta|2\rangle$ at a time $t_{1}$, where $|1\rangle$ and $|2\rangle$ are orthonormal states of the electron and $\alpha^{2} + \beta^{2}=1$.
Alice observes the electron at $t_{2}$ w.r.t. $|1\rangle$ and $|2\rangle$, and finds out that the electron is in $|1\rangle$.
So, Alice describes the electron system as follows:
$t_{1} \rightarrow t_{2} : \alpha|1\rangle + \beta|2\rangle \rightarrow |1\rangle$.
In fact, Alice and the electron (and whatever measurements devise Alice has) are all in a closed box, outside of which is another observer, Bob. Bob knows the initial condition of Alice at $t_{1}$ so he describes the evolution of the electron-Alice system as follows:
$t_{1} \rightarrow t_{2} : (\alpha|1\rangle + \beta|2\rangle) \otimes|Alice, initial\rangle \rightarrow \alpha|1\rangle \otimes|Alice1 \rangle + \beta|2\rangle \otimes|Alice2\rangle$,
where $|Alice1(2)\rangle$ denotes the state of Alice obtaining the measurement value 1(2). Then what is the state of the electron $alone$ from the perspective of Bob? Is it $\alpha|1\rangle\langle1| + \beta|2\rangle\langle2|$?
