## 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 $U$?

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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 – 2020-07-15T23:35:13.440

Actually, a simple example would be sufficient. – Rob – 2020-07-16T10:48:16.820

a simple example of what? – glS – 2020-07-16T11:28:43.663

An example of U, that would lead to a valid measurement. – Rob – 2020-07-16T13:39:54.693

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 – 2020-07-16T15:39:40.647

I am asking the following: Given the combined state $\rho_s \otimes \rho_a$, give an example (or a general form) of $U$, such that the quantity I wrote in my question, represents a valid measurement. – Rob – 2020-07-16T16:44:03.853