I have a few questions about measurement in Bell-state basis. In particular, if $Z = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}$ is for a measurement on the computational basis, then what is the representative matrix for a measurement in Bell-state basis.
I know that such a matrix can be constructed using spectral decomposition, but my Professor says the eigenvalues corresponding to 4 Bell-states remain unknown, so basically there is currently no physical quantity that helps on this kind of measurement.
However, Nielsen and Chuang (p.27) give a circuit for teleportation (basically Bell-basis measurement)
I wondered if $U^\dagger (Z\otimes Z)U$, where $U = (H\otimes I)CNOT$, is the needed matrix. It turns out that its eigenvectors are not Bell states. Can someone explain where I'm wrong here?