- #1

beefbrisket

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- 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: [itex]\rho_A[/itex]

Measurement instrument: [itex]\rho_B = |\phi\rangle\langle\phi|[/itex] (pure state)

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

After the projective measurement (and getting outcome [itex]a_k[/itex]), the total system state would be

[tex]\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|))}[/tex]

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 [itex]M_k \rho_A M_k^\dagger[/itex]? 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 [itex]U[/itex] being there.

System: [itex]\rho_A[/itex]

Measurement instrument: [itex]\rho_B = |\phi\rangle\langle\phi|[/itex] (pure state)

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

After the projective measurement (and getting outcome [itex]a_k[/itex]), the total system state would be

[tex]\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|))}[/tex]

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 [itex]M_k \rho_A M_k^\dagger[/itex]? 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 [itex]U[/itex] being there.