Taking a partial trace of a multipartite state for measurement

[beefbrisket]

TL;DR

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.