Do the probability amplitudes describe the probability of projective measurement?

I don't really know the difference between projective measurement and regular measurement or POVM. Until now, wherever I read about the subject I saw that the amplitude describes the probability of obtaining corresponding state when measuring it. I am not sure if this said measure is what you call a projective or regular. Can someone clear up the confusion? What's the about projective vs regular.

I saw somewhere that there is such a measurement M associated with POVM (corresponding to P??), but I am not sure what it says intuitively as well as mathematically. Which may as well help make clear the difference between the above mentioned ones.

What you call "regular" measurement is probably projective measurement onto the computational basis, $$|0\rangle$$ and $$|1\rangle$$. Most introductory quantum computing resources define a qbit as follows:

$$|\psi\rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} = \alpha|0\rangle + \beta|1\rangle$$

Where $$|\alpha|^2$$ gives you the probability of collapse to $$|0\rangle$$, and $$|\beta|^2$$ gives you the probability of collapse to $$|1\rangle$$. If this is what you know as "regular" measurement, then yes it's just projective measurement.

In addition to measuring in the computational basis, you can do a projective measurement onto any pair of orthogonal unit vectors. You do this by rewriting your quantum state in that basis. Here is a good video tutorial on how to do change-of-basis. For example, we can measure in the $$X$$ basis:

$$|+\rangle = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}$$, $$|-\rangle = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} \end{bmatrix}$$

Consider the following qbit value:

$$|\phi\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = 1|0\rangle + 0|1\rangle$$

This qbit value has a 100% chance of collapsing to $$|0\rangle$$ when measured in the computational basis. What if we were to measure it in the $$X$$ basis? First we have to do a change-of-basis to the $$X$$ basis, where it is written as:

$$|\phi\rangle = \frac{1}{\sqrt{2}}|+\rangle + \frac{1}{\sqrt{2}}|-\rangle$$

Note we did not change the value of the qbit at all, we are just writing it a different way (in a different basis). Now we see that if we measure $$|\phi\rangle$$ in the $$X$$ basis, it has a 50% chance of collapsing to $$|+\rangle$$ and a 50% chance of collapsing to $$|-\rangle$$.

It's called projective measurement because geometrically you're "projecting" your quantum state vector onto the measurement basis vectors, and the length of the projection on a basis vector gives you the probability of collapsing to that basis vector.

• But you just explained one sort of measurement which is the projective measurement. Though clearing a little bit up the confusion that doesn't answer my question completely. I believe I saw two different types of measurements . Maybe it has to do with POVM which I didn't write because I don't know what it means. According to your answer you just talk about the state collapsing to one of the basis states. Which of course is dependant on your basis choice. But doesn't help me much. Though I did upvoted your answer! – bilanush Feb 11 at 21:11
• Regular measurement is projective measurement. There are other measurement types like POVM, yes. What else do you want to know? Can you explain what you mean by "regular measurement" if the type of measurement I described is not regular measurement as you know it? – ahelwer Feb 11 at 22:16
• No, no. I am sorry . Your explanation was good and this is actually indeed the regular measurement I was probably reffering to. But you say they are the same thing. It's not that I knew two types. It's just that I heard there are two types . And it would help me greatly to know the second type. Might be it , the POVM one. Which I can't really understand even after reading Wikipedia. I mean, the regular measurement we were talking about , is simply understood as you described it. But what's POVM? – bilanush Feb 12 at 0:21
• I now tried to look it up, and I found it's some M measurement element associated with povm but I am not sure what's it. And how do you find this measurement M. Thanks for any help !!!! – bilanush Feb 12 at 0:21
• Okay! In that case could you edit your original question to be about POVM vs projective measurements? – ahelwer Feb 12 at 0:23