State of an entangled state after measurement

Question

I have this general state $\lvert \psi \rangle_{AB} = \sum_{\alpha\beta}c_{\alpha\beta}\lvert\alpha \rangle \otimes \lvert \beta\rangle$ of two qbits. I want to write the state of the system after the measurement. The measurement is performed only on the first qbit. Let's say that the value found was $\alpha_1$, I wrote:

$\lvert\psi\rangle_{AB} \longrightarrow \lvert \psi_{\alpha1}\rangle$ = $\lvert \alpha_1\rangle \langle \alpha_1 \rvert \otimes1\sum_{\alpha\beta}c_{\alpha\beta}\lvert \alpha\rangle\otimes\lvert \beta \rangle$ = $\lvert \alpha_1\rangle\otimes\sum_\beta\lvert\beta\rangle$.

Is it right? Is this the state of the system after measurement?

This is a decomposed state, as it seems to me. The lecture note I was studying said that the state of the system after the measurement is always a decomposable state, and I should show that this is true and also that the second qbit is dependent on the result of the measurement due to the first. I cannot see that. If the result of my calculation is right, the second qbit is the combination $\sum_\beta\lvert \beta\rangle$, but was eliminated any possibility of a tensor product with any other state other than $\lvert \alpha_1 \rangle$. That is, anything that is not $\lvert \alpha_1\rangle$ in the first qbit is not possible anymore, but any result in the second qbit is still possible.

What is wrong with my conclusion, and why is that?

Answer 1

You're right that the post-measurement state would be represented by the action of the projection operator $(\vert \alpha_1 \rangle\langle \alpha_1\vert \otimes \mathbb{I})$ on the pre-measurement state (up to a constant, of course). However, the resultant of this would not get rid of all the coefficients $c_{\alpha\beta}$. Rather (check this by explicitly writing the Kronecker delta function before you get rid of the summation over $\alpha$), it would give $\vert \alpha_1\rangle \otimes \sum_{\beta} c_{\alpha_1 \beta} \vert\beta\rangle$. Thus, the post-measurement state is a product state but the state of the $B$ subsystem is $\sum_{\beta} c_{\alpha_1 \beta} \vert\beta\rangle$ where the dependence of the coefficients on $\alpha_1$ makes it clear that the post-measurement state of the $B$ system depends on the particular state onto which the $A$ system collapses.

Notice that if $c_{\alpha\beta}$ were of the form $c_{\alpha} c_{\beta}$ then $\sum_{\beta}c_{\alpha_1\beta} = c_{\alpha_1}\sum_\beta c_{\beta}$ and thus the post-measurement state of system $B$ wouldn't depend on the outcome of the measurement on system $A$ because the prefactor $c_{\alpha_1}$ would be weeded out in normalization without any physical significance whatsoever. As you'd notice, this is simply reflecting the fact that the post-measurement state of one of the subsystems doesn't depend on the measurements done on the other subsystem in a product state but does depend on it in an entangled state.