Is there any operator in quantum mechanics that measure an observable with non-zero uncertainty? What does a measurement do?
The answer is: If the detector is designed to measure some observable O, it will leave the measured object, at least for an instant, in a zero-uncertainty state.
I want to know, in the context of quantum mechanics and base on Hilbert space, can define an operator to measure an observable with non-zero uncertainty?
 A: The uncertainty of an observable depends on the state $|\psi\rangle$ of the system that is being measured. The expectation value of the observable $A$ is given by
$$\langle A\rangle=\langle\psi|A|\psi\rangle$$
and the uncertainty is given by
$$(\Delta A)^2=\langle (A-\langle A\rangle)^2\rangle=\langle A^2\rangle-\langle A\rangle^2$$
Therefore, you can have a state with non-zero uncertainty in $A$ when $\langle \psi|(A-\langle A\rangle)^2|\psi\rangle\neq0$. There are many examples of systems like these. For example, a Gaussian wave packet has non-zero uncertainties in both position and momentum.
In terms of measurement, of course if you measure an observable the system is in an eigenstate of the observable and hence has $0$ uncertainly for that observable. But that does not make the new state a state of "non-zero uncertainty" in general, because the state could have uncertainty with respect to other observables. More explicitly, if we measure $A$ for our state so that now $|\psi\rangle=|a\rangle$, we now have $\Delta A=0$, but there could be (definitely is?) another observable $B$ such that $\Delta B\neq0$ for this new state $|a\rangle$.
If you want to make an observable where state after measuring $A$ has a non-zero uncertainty in $A$, then that is impossible. This is because the state after measuring $A$ has to be an eigenstate of $A$, so then $\Delta A=0$ after measurement always.
A: Defining an observable $M = \sum_m m P_m$, with $m$ the eigenvalues and $P_m$ the eigenspace projectors, is equivalent to defining a set of projective measurements $\{P_m\}$ such that $\sum_m P_m = I$ and $P_m P_{m'} = \delta_{mm'} P_m$
Since the operators $P_m$ are idempotent ($P_m^2 = P_m$), two consecutive measurements, one immediately after the other, necessarily yield the same result. Any other set of measurements $\{M_m\}$ would be measuring something else.
On the other hand, it is possible to define a set of measurements $\{M_m\}$ such that two immediately consecutive measurements can yield different results. This can be expressed using the POVM (positive-operator-valued measures) formalism. In that case, the measurement operators no longer are orthogonal projectors.
To give an example: such measurements can be used to unambiguously distinguish non-orthogonal quantum states. By virtue of the no-cloning theorem, this is impossible to achieve with complete reliability. However we can have a measurement that is sometimes inconclusive but never makes an error of mis-identification.
Let's take the simple case of a system prepared in one of two states $|\psi_1 \rangle = |0 \rangle$ or $| \psi_2 \rangle = \frac 1 {\sqrt 2}(|0 \rangle + |1 \rangle)$. We will then apply the following POVM:
$$\begin{align}
E_1 &= \frac {\sqrt 2} {1 + \sqrt 2} |1 \rangle \langle 1| \\
E_2 &= \frac {\sqrt 2} {1 + \sqrt 2} \frac {(|0 \rangle - |1 \rangle) (\langle 0| - \langle 1|)} 2 \\
E_3 &= \mathbf I - E_1 - E_2
\end{align}$$
We can see that if we have state $|\psi_1 \rangle$, there is zero probability of getting the result $E_1$: $\langle \psi_1 | E_1 | \psi_1 \rangle = 0$. Similarly, if we have state $| \psi_2 \rangle$, there is zero probability of getting the result $E_2$. Therefore, if we observe $E_1$, we know that the state was $| \psi_2 \rangle$ and vice versa.
However, in both cases, there is a non-zero probability of observing $E_3$, our "inconclusive" result:
$$\langle \psi_1 | E_3 | \psi_1 \rangle = \langle \psi_2 | E_3 | \psi_2 \rangle = \frac 1 {\sqrt 2} \approx 0.71$$
Adapted from:


*

*Quantum Computation and Quantum Information,  Nielsen and Chuang (2010)

*Unambiguous Quantum State Discrimination, Keyes (2005)

