I have always struggled to appreciate the measurement postulate (Section 2.2.3 of Nielsen and Chuang).
"Quantum Measurements are described by a collection of $\{M_m\}$ of measurement operators".
My problem/question concerns using matrices $M_m$ (as opposed to vectors) as measurement operators. Let me explain the confusion/problem with a couple of remarks.
The particular case of "Projective Measurements" (section 2.2.5) is very intuitive -- they represent "measuring" the quantum state on an orthonormal basis. A generalization of projective measurements should measure the quantum state on a non-orthonormal basis (i.e., a set of vectors that are not orthogonal to each other). Why is that not a sufficient or meaningful generalization? (I realize that perhaps the postulate is even more general?).
Outcome of measuring a state $|\psi\rangle$ over/using an operator/observable $A$ (e.g., Pauli matrices) is an eigenvector of $A$. Then, in the measurement postulate, why is the outcome not an eigenvector of one of the operators $M_m$ rather than $M_m |\psi\rangle/c$ (here, $c$ is the normalization factor)?
What is the measurement operators' physical (or intuitive) meaning?
Thanks.
