Measurement operators: singularity and other restrictions?
A posulatuate used in <PERSON> & <PERSON>'s Quantum Computation and Quantum Information:
Quantum measurements are described by a collection ${M_m }$ of measurement operators. These are operators acting on the state space of the system being measured. The index $m$ refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is $|\psi\rangle$ immediately before the measurement then the probability that result $m$ occurs is: $$p(m) = \langle\psi| M_m^{\dagger} M_m|\psi\rangle $$
and the state of the system after the measurement is $$ \frac{M_m|\psi\rangle}{\sqrt{p(m)}} $$
As the $p(m)$ must sum to one, the measurement operators must satisfy the completeness equation, $$\sum M_m^{\dagger} M_m = I$$
No other restraints are noted for these measurement operators.
The operators are not explicitly required to be singular, which caught my attention.
Let's say I have a set of ${M_m }$ measurement operators which describe some measurement, $M$. For some other measurement, $N$, I have a set of ${N_n }$ measurement operators describing it. 680
It so happens that the operator $N_b = M^{-1}_a$ for a specific $b$ and $a$.
Upon making my first measurement, $M$, on an ensemble of states all in $|\psi\rangle$, some fraction, $p(a)$, of them end up in the state: $$ |\psi^{'}\rangle = \frac{M_a|\psi\rangle}{\sqrt{p(a)}} $$ On these states I choose to make my measurement $N$. Some fraction, $p(b)$, of them end up in the state: $$|\psi^{''}\rangle = \frac{N_b|\psi^{'}\rangle}{\sqrt{p(b)}} $$ $$ = \frac{N_b M_a |\psi\rangle}{\sqrt{p(b)p(a)}} = \frac{M^{-1}_a M_a |\psi\rangle}{\sqrt{p(b)p(a)}} = \frac{|\psi\rangle}{\sqrt{p(b)p(a)}}$$
$$ |\psi^{''}\rangle = \frac{|\psi\rangle}{\sqrt{p(b)p(a)}} $$
This would be an interesting result, suggesting that if I perform two measurements some fraction of the time, $p(a)p(b)$, I could return to the initial state and have obtained the knowledge of $p(a)$ and $p(b)$ along the way. Of course, since the division by $\sqrt{p(a \text{ or } b)}$ is supposed to serve as normalization term, the final result suggests that this process only works in the trivial case $p(a) = p(b) = 1$. I.e. $M_{a} = N_{b} = I$.
By contradiction (I guess?) it's shown that measurement operators ${ M_{m} }$ must all be singular. Or, at least, the inverse of any $M_m$ cannot be a member of any other set of measurement operators.
Had I been careful, I imagine this result could have been found by looking at the completeness equation directly.
Question: how could the singularity of all ${ M_{m} }$ be shown from the completeness relation and, further, are there any authors who go into the limitations on ${ M_{m} }$? 680