Taking a partial trace of a multipartite state for measurement

[beefbrisket]

TL;DR Summary

Seeking help in algebraic manipulation of a partial trace on a multipartite state to get it into the form of operators operating on only one of the subsystems' state.

I am attempting to understand how POVMs fit in with quantum measurement, and I think I am getting tripped up in notation when it comes to multipartite systems. The situation is as follows:

System: $\rho_A$
Measurement instrument: $\rho_B = |\phi\rangle\langle\phi|$ (pure state)

The multipartite system starts in state $\rho_{AB} = \rho_A \otimes \rho_B$ and to make a measurement the instrument interacts with the system resulting in a state $U \rho_{AB} U^\dagger$. Then a projective measurement is taken on the instrument only. The operator corresponding to eigenvalue $a_k$ for that would be $1 \otimes |k\rangle\langle k|$. Let's assume no degeneracy in eigenvalues.

After the projective measurement (and getting outcome $a_k$), the total system state would be
$$\frac{(1 \otimes |k\rangle\langle k|)U \rho_{AB} U^\dagger(1 \otimes |k\rangle\langle k|)}{tr(U \rho_{AB} U^\dagger(1 \otimes |k\rangle\langle k|))}$$

Focusing on the numerator, I take a partial trace wrt system B to find the state of system A. After doing so, how can I manipulate it into the form $M_k \rho_A M_k^\dagger$? I think I am getting bogged down in the Dirac notation somehow. I'm unsure how to apply the basis elements with which I am taking the partial trace due to $U$ being there.

 

[A. Neumaier]

You need to note that expressions such as $\langle k| U|\phi\rangle$ are still operators on the system.

To practice, you could temporarily work in finite dimensions and represent the tensor product as a Kronecker product, then ordinary matrix calculus with indices works. At the end, translate your derivation back into bra-ket notation.