Povm's and Von Neumann Meaurements

[bhobba]

Hi All

Read a thread that about Von-Neumann observations that was closed because it was a bit too vague, but I sort of got a sense of what the poster was on about - and it also is interesting anyway for anyone that doesn't know it so I thought I would do a post about it.

Since Von-Neumann's time a lot of work has been done on measurements eg:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

It turns out that what Von-Neumann wrote in his classic text on QM about measurements has now been generalized from Von-Neumann measurements to what are called POVM's.

Here is a summary.

First we need to define a Positive Operator Value Measure (POVM). A POVM is a set of positive operators Ei ∑ Ei =1 from, for the purposes of QM, an assumed complex vector space.

The basic measurement postulute of QM is an observation/measurement with possible outcomes i = 1, 2, 3 ... is described by a POVM Ei such that the probability of outcome i is determined by Ei.

In fact from that alone and non-contextuality (ie the probability is not affected by what POVM it is part of) you can derive the Born Rule.

This is observation in the form of its modern generality. Von-Neumann used a subset of this and to understand it we need to define what is called a resolution of the identity, which is POVM that is disjoint ie if you take any two elements of the POVM A and B then AB=0. Such are called Von-Neumann observations. We know from the Spectral theorem Hermitian operators, H, can be uniquely decomposed into resolutions of the identity H = ∑ yi Ei. So what we do is given any observation based on a resolution of the identity Ei we can associate a real number yi with each outcome and uniquely define a Hermitian operator O = ∑ yi Ei, called the observable of the observation.

But as the link I gave shows, by use of a probe to observe something then observing the probe you are forced to generalize Von-Neumann measurements to POVM's so that is the modern view of a measurement. Of course once you do that you can't always form an operator of the measurement because it may no longer be a resolution of the identity.

Thanks
Bill

 

[bluecap]

Hi All

Read a thread that about Von-Neumann observations that was closed because it was a bit too vague, but I sort of got a sense of what the poster was on about - and it also is interesting anyway for anyone that doesn't know it so I thought I would do a post about it.

Since Von-Neumann's time a lot of work has been done on measurements eg:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

It turns out that what Von-Neumann wrote in his classic text on QM about measurements has now been generalized from Von-Neumann measurements to what are called POVM's.

Here is a summary.

First we need to define a Positive Operator Value Measure (POVM). A POVM is a set of positive operators Ei ∑ Ei =1 from, for the purposes of QM, an assumed complex vector space.

The basic measurement postulute of QM is an observation/measurement with possible outcomes i = 1, 2, 3 ... is described by a POVM Ei such that the probability of outcome i is determined by Ei.

In fact from that alone and non-contextuality (ie the probability is not affected by what POVM it is part of) you can derive the Born Rule.

This is observation in the form of its modern generality. Von-Neumann used a subset of this and to understand it we need to define what is called a resolution of the identity, which is POVM that is disjoint ie if you take any two elements of the POVM A and B then A-B=0. Such are called Von-Neumann observations. We know from the Spectral theorem Hermitian operators, H, can be uniquely decomposed into resolutions of the identity H = ∑ yi Ei. So what we do is given any observation based on a resolution of the identity Ei we can associate a real number yi with each outcome and uniquely define a Hermitian operator O = ∑ yi Ei, called the observable of the observation.

But as the link I gave shows, by using of a probe to observe something then observing the probe you are forced to generalize Von-Neumann measurements to POVM's so that is the modern view of a measurement. Of course once you do that you can't always form an operator of the measurement because it may no longer be a resolution of the identity.

Thanks
Bill

Bill. Please ask your physicist friends how familiar they are with the von Neumann measurements stuff. Or based on your understanding, how many practicing physicists are familiar with it. I'd like to know if this is part of undergraduate or Ph.D. subjects so whenever I meet other physicists I have idea whether von Neumann is a normal subject or an exotic one (that normal students don't need to bother with). Remember many practicing physicists with Ph.D. don't care about Bell's Theorem for example because it wouldn't make their condense physics calculations better. so I wonder if the normal physicists don't care about the von Neumann measurements stuff too.. because if this is the case.. then at least I need to first introduce about the von Neumann thing before talking to them about it and not assuming they are fully familiar with it. Thanks.

 

[bhobba]

Bill. Please ask your physicist friends how familiar they are with the von Neumann measurements stuff. Or based on your understanding, how many practicing physicists are familiar with it. I'd like to know if this is part of undergraduate or Ph.D. subjects so whenever I meet other physicists I have idea whether von Neumann is a normal subject or an exotic one (that normal students don't need to bother with). Remember many practicing physicists with Ph.D. don't care about Bell's Theorem for example because it wouldn't make their condense physics calculations better. so I wonder if the normal physicists don't care about the von Neumann measurements stuff too.. because if this is the case.. then at least I need to first introduce about the von Neumann thing before talking to them about it and not assuming they are fully familiar with it. Thanks.

My physicist friends are all here - I am retired these days and was a programmer before that.

After you have studied to the level of say Sakuari - Modern QM there are all sorts of directions you can go. You can learn even more advanced ordinary QM from Ballentine, you can go into QFT, you can go into quantum foundations, you can look into deep mathematical issues like Rigged Hilbert Spaces - the directions it can take you are pretty well endless. Which ones decided on looking into the modern theory of quantum observations - I have zero idea. I looked into it as part of understanding QM better after delving into the mathematical issues - but that's just me.

It can be studied as an advanced undergraduate or as a graduate student.

Thanks
Bill

 

[kith]

All physicists are familiar with von Neumann measurements. They might not call it by this name but many textbooks talk about von Neumann measurements exclusively.

The more general POVM approach is what many physicists are unfamiliar with.

 

[vanhees71]

I think the reason, why many physicists (including myself) are not very familiar with the concept of "weak measurements", based on the POVM (positive operator-valued measure) formalism is, that it is a quite new concept in the context of the foundations of quantum theory. The reason, why it's useful today is simply the technological progress which enables to do experiments which have been impossible a few years ago. It's just progress of science! I think, it helps us to understand QT better and better.

On the other hand, what hinders the understanding of QT nowadays is the opposite extreme of letting more and more philosophical gibberish enter the scientific literature, which in my opinion is as bad as the suppression of such ideas in some decades before, because it distracts from a real understanding in the sense of well-justified scientific methodology, which finally always rests on empirically testable phenomenology and not "scholastic speculations".

In the decades after WW II (50ies-late 60ies, i.e., before Bell's work) it was almost career destroying for a young physicist to deal with the foundational questions at all. In some way this was justified, because many of the foundational questions were not formulated in a scientific way, i.e., it was not making testable predictions to decide empirically, which point of view on a subject like the socalled "measurement problem" or the "inseparability of far distant parts of quantum systems" in terms of entanglement is right or wrong, and then indeed it's not accessible to the scientific method and thus doesn't belong to physics. On the other hand, theoretical work on finding ways to decide these questions, which are of course in a way important for a full understanding of the meaning of QT, has proven very profound and even lead to new technology like quantum cryptography and may lead to more developments like quantum computing. This changed with Bell, who with his inequalities indeed made a purely philosophical question about the inseparability of quantum systems (raised by the famous EPR paper) assesible to empirical tests, and today it's among the best tested facts about QT that its predictions of far-distant correlations stronger than possible in any local determinstic theory, is indeed correct, making the above mentioned technological progress possible as a spin-off.