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Are all measurement operators on a quantum mechanical system defined by a Hilbert space, such that all possible post-measurement states are orthogonal? For example measuring a qubit in some orthonormal basis $\{|0\rangle,|1\rangle\}$. The possible outcome states after measurement are $|0\rangle$ and $|1\rangle$. I know the example I gave above is projective measurement, a special case of general measurement. So is there an example where all possible post-measurement states are not orthogonal?


I know if the measurement operators are $\{M_m\}$ ( $m$ denotes a possible outcome) then if the outcome is $m$ post-measurement state is $\frac{M|\psi \rangle}{|M|\psi \rangle|}$ ( $|\psi\rangle$ being initial state of system ) such that $\sum M^{\dagger}_m M_m=I$. Thus I can mathematically see that the example I am looking for possible , but I can't come up with one having some physical significance.

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Yes. Nonweak measurements correspond to Hermitian (or self adjoint) operators. The results are 1) an eigenvalue and 2) you project the state vector onto the corresponding eigenspace.

The projections onto different eigenspaces produce eigenvectors with different eigenvalues, and eigenvectors of a symmetric operator with different eigenvalues are orthogonal.

So for nonweak measurements the different outcomes are orthogonal. Always.

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