# 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 $\rho'$.

• Got too late to this one. Craig's answer touches on the main issues, and Emilio's comment dots the i. You can do this only with post-selection and in the ensemble sense by measuring along the direction of the (common) eigenstates. – udrv Jun 23 '16 at 18:20

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.

• You can measure and then decide randomly, using coins of different bias for the two possible outcomes, whether to keep the result. That will increase the purity but not to 100%. – Emilio Pisanty Jun 23 '16 at 17:53
• @EmilioPisanty That will work if you replace 'whether to keep the result' with 'whether to post-select'. Measurement will increase the purity, but not necessarily maintain the direction (my interpretation of the question is that they also want to maintain the direction). – Craig Gidney Jun 23 '16 at 17:57
• Thanks for the reply. Yes, directions are fixed. We can imagine the situation on the Bloch sphere. The required job is to change the length of Bloch vector while $\theta$ and $\phi$ are fixed. Proper measurements operator can change the length of Bloch vector without changing directions but I don't know the procedure how to choose appropriate measurement operator. – Parveen Kumar Jun 23 '16 at 19:23
• @ParveenKumar What kind of measurement operator do you want? An axis? An observable (Hermitian matrix)? A POVM? – Craig Gidney Jun 23 '16 at 20:23
• Hermitian matrix won't work because Hermitian matrix provides only rotation on the Bloch sphere. They can't change radial vector. They changes only $\theta$ and $\phi$. Probably POVM is required to increase the radial vector without affecting the angles. – Parveen Kumar Jun 24 '16 at 8:47