How to choose proper measurement operator? Let's assume I have two states inside the Bloch sphere, at radial vectors $r_1$ and $r_2$ respectively $(r_1<r_2<1)$. Their angular location is same. These are like:
\begin{equation} 
\rho = 
\begin{pmatrix}
\frac{1+r_1 \cos\theta}{2} &\frac{r_1 \exp(-i\phi)\sin\theta}{2} \\
\frac{r_1 \exp(i\phi)\sin\theta}{2} &\frac{1-r_1 \cos\theta}{2} \nonumber
\end{pmatrix}
\end{equation}
and another state as
\begin{equation} 
\rho' = 
\begin{pmatrix}
\frac{1+r_2 \cos\theta}{2} &\frac{r_2 \exp(-i\phi)\sin\theta}{2} \\
\frac{r_2 \exp(i\phi)\sin\theta}{2} &\frac{1-r_2 \cos\theta}{2} \nonumber
\end{pmatrix}
\end{equation}
I know that measurement can increase radial vectors but I don't know the procedure to find the suitable measurement operators which can relate the above states $\rho$ and $\rho'$.
 A: When don't you condition on the result, measurement of a qubit can only decrease its purity (you end up with less information than you started with).
When you do condition on the result, measurement of a qubit will make it 100% pure but there are two possible results. One possible result is along the measurement axis you measured. The other is against the measurement axis (well... along it, but negative-ward instead of positive-ward). This will invert the angles instead of only changing the radius.
So I'm not sure how you could do what you want (increase radius without affecting angles) without post-selecting. Measure the qubit along the axis component of its Bloch sphere representation, retry the whole experiment up to that point until you get the "along" answer instead of the "against" answer, then continue.
Here's an example circuit, where I setup a qubit to have a mixed state pointing towards the -X state then post-select it into the -X state. But keep in mind that it's the post-selection that's entirely responsible for the final state; states along the Z axis would also end up at -X in this case.

If you want to decrease the radius, that's easy. use the qubit to control small/misaligned rotations on other qubits (that's what I do in the above circuit). Partial measurements also work, as long as you don't condition on the result.
