# Representing a von Neumann measurement as $[\mathcal{I} \otimes P_i] U(\rho_s \otimes \rho_a)U^{-1} [\mathcal{I} \otimes P_i]$, how do we choose $U$?

Given the state of a system as $$\rho_s$$ and that of the ancilla (pointer) as $$\rho_a$$, the Von-Neumann measurement involves entangling a system with ancilla and then performing a projective measurement on the ancilla. This is often represented as $$[\mathcal{I} \otimes P_i] U(\rho_s \otimes \rho_a)U^{-1} [\mathcal{I} \otimes P_i],$$ where $$\mathcal{I}$$ is the identity on system space, $$P_i$$ is the projector corresponding to $$i$$-th outcome, and $$U$$ is the combined unitary.

My question: How to choose the form of $$U$$?

• are you asking given a map $\Phi$ represented as $\Phi(\rho)=\operatorname{tr}_a[U(\rho\otimes \rho_a)U^\dagger]$, how to find the unitary $U$ in this representation?
– glS
Jul 15 '20 at 23:35
• Actually, a simple example would be sufficient.
– Rob
Jul 16 '20 at 10:48
• a simple example of what?
– glS
Jul 16 '20 at 11:28
• An example of U, that would lead to a valid measurement.
– Rob
Jul 16 '20 at 13:39
• I still don't know if I understand what you are asking. Every map can be represented in this form. But you are asking about a von Neuman measurement, not a map. By "von Neumann measurement" you mean a POVM, or more specifically a projective measurement? Then you are essentially asking what is the unitary representation of maps that represent measurements?
– glS
Jul 16 '20 at 15:39