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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 $U$?

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  • $\begingroup$ 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? $\endgroup$
    – glS
    Jul 15 '20 at 23:35
  • $\begingroup$ Actually, a simple example would be sufficient. $\endgroup$
    – Rob
    Jul 16 '20 at 10:48
  • $\begingroup$ a simple example of what? $\endgroup$
    – glS
    Jul 16 '20 at 11:28
  • $\begingroup$ An example of U, that would lead to a valid measurement. $\endgroup$
    – Rob
    Jul 16 '20 at 13:39
  • $\begingroup$ 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? $\endgroup$
    – glS
    Jul 16 '20 at 15:39

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