Why are POVMs useful? Are they just an axiomatic way to define measurement?

Question

I know the definition of projective measurement, generalized measurement, POVM.

I understand the usage of generalized measurement for the reason that it can model experiments "easier" (for example measurement of a photon that will be destructive so that measuring again the state just after the first measurement will give me another answer).

However, I am still kinda confused by why we have introduced the notion of P.O.V.M. For me we have everything we want from generalized & projective measurement.

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Would you agree with me if I say that POVM is just an axiomatic way to define statistics of measurement. There is nothing much to understand/overthink.

In the sense, we ask the minimal mathematical properties that our measurement operator must fullfil with respect to statistical behavior which is:

• they are semidefinite positive (to have positive probabilities)
• they sum up to identity (to have probability summing up to $1$)

and we relate our measurement operator with the physics:

$$p(m)=\mathrm{Tr}(E_m \rho)$$ where $m$ is the outcome, $E_m$ the associated POVM.

Answer 1

If you want to prove a no-go theorem in quantum mechanics, such as the impossibility of distinguishing two non-orthogonal states given a single unknown state, you want to prove that even if you add ancillae, apply unitaries, do projective measurements in weird bases, apply unitaries conditional on previous measurements, etc, you still can't distinguish two non-orthogonal states. One nice way to do that is instead of considering all possible combinations of operations you might perform on your system, you realize that every possible sequence of operations is really a POVM, and every POVM can be realized by some convoluted combination of elementary operations. Then you just prove that no POVM can distinguish two non-orthogonal states, and you're done!

Answer 2

For me, generalised measurements cover everything (obviously, that's why they're generalised), with projective measurements being a simple case that covers what we usually want to be doing.

So, yes, why introduce POVMs which are basically the generalised measurements but without the output state? Because they describe what actually happens in some experiments. If you're using optics, for example, where measurement of a photon is destructive, i.e. you do not have the photon afterwards, the POVMs perfectly describe what's going on.

What I think is a reasonable follow-up question is why so many courses/texts etc put as much weight as they do on POVMs. I believe that the reason is simply that, when performing many quantum information protocols, we don't care about the state after measurement, we only care about the probabilities of the different outcomes. For example, if you're trying to identify what state you have, you only care about which measurement result you get, not the final state. Then it's a mathematical convenience that you're dealing with $E_i$ instead of having to calculate $M_i^\dagger M_i$ first.

Answer 3

POVMs model the most general way to measure something about a quantum state. I'd argue POVMs differ from "generalized measurements" in that they are tools built to answer different questions.

A POVM tells you about how quantum states are converted in probability distributions. The focus is on that measurement probabilities, nothing more and nothing less.

Generalized measurements also deal with a possible post-measurement state. Aside from the fact that in many situations you do not have a post-measurement state, the fact of the matter is that you might not care about post-measurement states for whatever you are studying. If you do, then it makes sense to use generalized measurements, or equivalently, the corresponding quantum-to-classical channels (which codify the same identical information as generalized measurements anyway). You often don't, and in these cases it's therefore easier/preferable to work directly with POVMs.