Talk:POVM

Analogy

The article makes the analogy

POVM is to projective measurement what the density matrix is to the pure state.

I don't think this is quite right or at least I don't clearly understand what the analogy means since projective measurements can result in proper mixed states (even starting from pure states).

--CSTAR 17:10, 22 January 2006 (UTC)

I agree this sentence is confusing. The analogy is this:

1st try: Density matrixes have the ability to describe part of a larger system that is in a pure state. POVMs have the ability to describe the action of PVM in a larger space on the part of a state in a subspace.

2nd try: Unlike the pure state formalism, the density matrix formalism is able to always completely describe the state of part of a larger system. Unlike projective measurement formalism, a POVM is always able to describe the action on a state contained within the subspace of a measurement in a subspace of the space of the measurement.

3rd try: PVMs in a space are described by POVMs in a subspace of that space. Pure states in a space are described by density states in a subspace of that space.

Feel free to change the sentence if you can express this idea more clearly.

Anyways, I always think of POVMs in terms of their similarities and differences to PVMs. I was thinking of restructuring the article to introduce the definition and properties of POVMS in a table with corresponding entries for PVMs.--J S Lundeen 04:04, 27 January 2006 (UTC)

Ancillas

Mct Mht, I am unclear what you mean about Naimark's theorem not being applied here. Whenever you bring in an ancilla you extend the Hilbert space (i.e. to N dimensions). The coupling that you mentioned can be folded into the projective measurement on the extended hilbert space. In conclusion, Naimark's theorem prescribes the general strategy for performing a POVM. Coupling to ancillas is not an exception.--J S Lundeen 21:58, 15 June 2006 (UTC)

actually, that's not quite right. in the finite dimensional case, it's somewhat trivial either way. but Naimark's dilation theorem says the measure space on which you define the POVM is fixed. when you couple to the system an ancilla, that's no longer true, and the problem becomes finding the unitary dilation of an isometry. that's, in general, less deep than Naimark's theorem. Mct mht 23:03, 15 June 2006 (UTC)

i think invoking Naimark's theorem, when talking about the finite dimensional case, is somewhat misleading by itself, as Naimark's result is much deeper than that and a PVM (without ancilla) can be found without recourse to Naimark's theorem at all. the fact that it is in Peres's book not withstanding. Mct mht 01:09, 16 June 2006 (UTC)

Okay, I will take a look Peres' book. However, 'Neumark's dilation theorem states that measuring a POVM consisting of a set of n>N operators acting on a N-dimensional Hilbert space can always be achieved by performing a projective measurement on a Hilbert space of dimension n then consider(ing) the reduced state.' appears to include the follow procedure: 'In practice, however, obtaining a suitable projection-valued measure from a given POVM is usually done by coupling to the original system an ancilla.'

So although Naimark's theorem may not be necessary, it is sufficient.--J S Lundeen 10:38, 16 June 2006 (UTC)

If the "coupling to an ancilla" doesn't increase the number of effects of the POVM, i.e. the PVM has n number of elements, then yeah sure that's what Naimark's theorem says. But it is also just linear algebra. It is, IMHO, highly misleading and an injustice to Naimark's result. Compare with the situation where the number of Borel sets, therefore the elements of the POVM, is not finite, it is more appropriate then to use Naimark's. Mct mht 16:14, 16 June 2006 (UTC)

Section: Quantum properties of measurements

The initial paragraph of this section reads like a paper abstract, even going so far as using "we show". 136.186.9.141 (talk) 04:13, 16 December 2010 (UTC)

Yes, its as clear as mud. It needs a total re-write. Some contructive criticism:

• what is Pi-hat? why does it have a subscript n? What does the hat mean? What,s Pi without that hat?
• what is Theta-hat? Why the subscript m? what is Theta without the hat?
• How can a pi be a 'projectivity'? a trace is a scalar unless its a partial trace .. is pi a scalar?

User:Linas (talk) 20:57, 23 November 2013 (UTC)

real vs. complex

Comparingdefnitions in this article to the PVM article, I see, in this article:

for every ξ $\in$ H,

$E\mapsto \langle F(E)\xi \mid \xi \rangle$

is a non-negative countably additive measure on the σ-algebra M.

whereas in PVM, there's this:

for every ξ, η ∈ H, the set-function

$A\mapsto \langle \pi (A)\xi \mid \eta \rangle$

is a complex measure on M

I'm getting hung up on two things: 1) for the POVM case, the measure implicitly real-valued (it has to be, cause F is hermitian); but that should be said directly. 2) The PVM definition uses two vectors ξ, η .. this is equivalent to using one vector, giving a real-valued measure, but its disconcerting, since its not immediately obvious that the measure is complex due to using two vetors, instead of it being complex for some other reason... this should be clarified. User:Linas (talk) 17:50, 23 November 2013 (UTC)

They are the same, by the polarization identity of the inner product. Mct mht (talk) 12:24, 26 November 2013 (UTC)

I added a remark that makes this clear. 89.217.0.87 (talk) 14:49, 3 May 2015 (UTC)

Schizophrenic content

This article seems to be talking about two different things. The general definition, in the first section, talks about a measurable space (X,M) and its fibration with Hilbert spaces as fibers. This is fine, and is fully consistent with the PVM article. Then the rest of the article seems to assume that X is a single point: X is never mentioned again, the sigma algebra is never mentioned again. Instead, it seems to talk only about having an over-complete set of operators on a single Hilbert space. What's more, it always seems to implcitly assume its finite dimensional. The concluding section demos a lift of 2D to 3D!! Yes, I understand that, due to the direct integral treatment, as given in the PVM article, we could treat the whole thing as a single Hilbert space. The disappearance of the concept of measure right after the defintion is disconcerting. It doesn't seem that most of the contributors to this article are even aware of this. By contrast, clicking over to Naimark's dilation theorem, one gets a stand-alone definition of a POVM that is in sync with the PVM article, but is entirely absent from this article.

Anyway, the last comprehensible version of this article seems to be this from July 2009. After that, User:Tercer made changes that damaged the article, and it accumlated cruft and confusion evermore. Strongly tempted to revert all edits from the past 4 years ... can someone please fix this mess? User:Linas (talk) 21:53, 23 November 2013 (UTC)

Damaged the article? I'm afraid you don't know what are you talking about. I merely added some remarks to explain why a POVM does not uniquely define the post-measurement state. I guess the issue is that the article is a mixture between the mathematician's point of you (which is probably yours), that cares about measure theory, and the physicist point of view, who cares about measurement in quantum mechanics. Mateus Araújo (talk) 08:49, 26 November 2013 (UTC)

That the article being a mixture is OK, possibly even great if done right. But it's not in this case. Disjointed-ness in the content as Linas pointed out (good to see you, Linas) is not to be defended. Operator theory proper is probably a reach for most physicists but article can be made coherent with input from both sides and dialogue/mutual checking. Mct mht (talk) 12:33, 26 November 2013 (UTC)

I agree, the article needs a lot of work. 89.217.0.87 (talk) 14:41, 3 May 2015 (UTC)

Possible non-consensus claim

From the section entitled "Quantum properties of measurements":

A recent work by T. Amri[1] makes the claim that the properties of a measurement are not revealed by the POVM element corresponding to the measurement, but by its pre-measurement state. This one is the main tool of the retrodictive approach of quantum physics in which we make predictions about state preparations leading to a measurement result...

The whole section appears to be a presentation of a possible non-consensus (or even fringe) claim about where the causation comes from in quantum mechanics.

The claim may be too broad for this article. It is not about the POVM operators, but rather, employs them to make an argument whose real point is about the interpretation of quantum mechanics, I think. On this basis, the material should probably be housed somewhere else.

The second half of the section introduces "nonclassicality" but it doesn't appear to say much about it.

Finally, the section is almost incoherent from a technical point of view, because it introduces a lot of variables without defining them. The equations appear to be lifted straight from a paper?

I would contrast this section with the previous one, "Neumark's dilation theorem" and its subsection "Post-measurement state", which are very poorly written, but appear to have important content.

I would recommend moving the section entitled "Quantum properties of measurements" to the talk page or deleting it.

89.217.0.87 (talk) 15:12, 3 May 2015 (UTC)

The assumptions for the very interesting example need more work

The section entitled "An example: Unambiguous quantum state discrimination" is much better written and more interesting than the previous two sections.

Still, there is something I don't understand.

In order to justify the "probabilities" discussed in the example, I made explicit the assumption that the unknown prepared states were drawn with equal probability from the permitted set.

However, this is nowhere to be found in the original article, and possibly it is a gratuitous assumption that is not part of the conceptualization of the problem. Is it really necessary?

Clearly, the quantum measurements yield probabilities, but they are conditioned on the input state and the measurement chosen, so to get an overall distribution, it might appear that both the input choice and the measurement choice need an a-priori probability distribution.

But it is possible to get probabilities out of the problem in another way. It could also be that the correct formulation is somewhat different, namely that the input states are not chosen with equal probability, but instead are chosen by a malicious opponent whose aim is to reduce the percentage of UQSD.

Then the probabilities of 50-50 and 50-50 that are cited in the article would be explained as a minimax strategy between the opponents.

It can be put as follows. Let i be a variable that indicates all possible inputs. Let A be the probability distribution of the inputs (under opponent control). Let B is the probability distribution of the outputs (under our control, and we are allowed to use a mixed strategy of any possible PVM's), then the 25% is the result of

max_B min_A P(UQSD | A,B).

But by theorems about zero-sum games, this is equal to the simpler expression

max_B min_i P(UQSD | i,B),

which no longer mentions an input probability distribution, but simply says we're doing the best we can do without knowing the input. This would allow us to remove the reference to a probability distribution on the inputs.

But it would have a cost: the derivation would become more complicated than the one already presented in the article. In the article derivation, the opponent's optimal strategy of 50/50 between Φ and Ψ is plugged in without further ado, to leap directly to the mixed probability. This was already true before I made it into an explicit assumption. If the assumption about the opponent's most damaging strategy is dropped, then it must be derived, or at least a nod made in this direction.

178.38.119.178 (talk) 17:57, 3 May 2015 (UTC)

Polarizers and number of modes at the end of the very interesting example

For a specific example, take a stream of photons, each of which is polarized along either the horizontal direction or at 45 degrees. On average there are equal numbers of horizontal and 45 degree photons. The projective strategy corresponds to passing the photons through a polarizer in either the vertical direction or -45 degree direction. If the photon passes through the vertical polarizer it must have been at 45 degrees and vice versa. The success probability is 25% . The POVM strategy for this example is more complicated and requires another optical mode (known as an ancilla). It has a success probability of 29.3%.

It would be nice to see this spelled out explicitly.

(1) Are there 2 or 4 optical modes (in total) available for polarized light? I think we need 4 to make it work (actually 3, but they surely come in pairs). Are the other two modes circularly polarized, or something like that?

(2) Can we describe the improved setup in concrete terms, such as inserting a "circular polarizer" or "1/4 wave plate"?

178.38.119.178 (talk) 20:13, 3 May 2015 (UTC)

Probability (density)

@Tercer, I noticed you reverted the edit by @Nathanielvirgo, stating:

In the continuous case what you get is a probability density, not a probability. You still get outcomes, though, not "events"

However, in the continuous case, "the" probability given by:

${\text{Prob}}(i)=\operatorname {tr} (|\psi \rangle \langle \psi |F_{i})=\langle \psi |F_{i}|\psi \rangle$

makes little sense to me. I think the easiest solution would be to remove the word "(density)" . Roffaduft (talk) 09:20, 11 October 2024 (UTC)

What's wrong with it? The probability density of observing a particle at position x is given by $\operatorname {tr} (|\psi \rangle \langle \psi ||x\rangle \langle x|)=|\langle x|\psi \rangle |^{2}$. Tercer (talk) 09:28, 11 October 2024 (UTC)

If a particle adheres to a (continuous) probability density function then, by definition, the probability that a particle will be at an exact position $x$ is zero. Using the Expectation value (quantum mechanics) would be more appropriate in this case. Roffaduft (talk) 09:36, 11 October 2024 (UTC)

That's why the formula gives you the probability density, not the probability. Tercer (talk) 21:38, 11 October 2024 (UTC)

What are you talking about? It literally says:

$\langle F(E)\psi \mid \psi \rangle$ can be interpreted as the probability (density) of outcome $E$ ... That is, .. probability of obtaining it .. is given by

${\text{Prob}}(i)=\operatorname {tr} (|\psi \rangle \langle \psi |F_{i})=\langle \psi |F_{i}|\psi \rangle$

So what is it? A probabilty density, or a probability? Roffaduft (talk) 03:35, 12 October 2024 (UTC)

It's a probability, in the discrete case, and a probability density, in the continuous case. Tercer (talk) 07:29, 12 October 2024 (UTC)

I fully agree that the statement makes sense in the discrete case, but if $\langle \psi |F_{i}|\psi \rangle$ were to be continuous, it no longer does. That's why I suggested to simply remove "(density)" as it then becomes clear from the context that only the discrete case is being discussed. Alternatively, one could add a mathematical description for the continuous case.

"The Born rule and its interpretation" gives a very nice overview on the correct use of mathematical terminology. Roffaduft (talk) 07:41, 12 October 2024 (UTC)

It's a measure, not a density function. Measures assign probabilities to events even in the continuous case. To talk about densities in a measure theoretic context such as this you would have to use some analog of the Radon-Nikodym theorem. Perhaps this can be done and maybe it's standard, but it's not explained in the article and so jumping to talking about densities at this point is a non sequitur. Nathaniel Virgo (talk) 18:15, 11 October 2024 (UTC)

In any case, the measure theoretic version is more general than a density function version would be, and it's what the article is about. Nathaniel Virgo (talk) 18:26, 11 October 2024 (UTC)

This is just a classic example of the area of tension between physicists and mathematicians. The former has a tendency to introduce terminology with little regard for its mathematical origin. E.g.:

• Eigenstates and pure states (as opposed to eigenfunctions and pure points)
• Density operator
• Born rule "the probability density of finding a system in a given state" (say what?)

Trying to interpret the latter two while adhering to classical probability theoretic constructs (e.g. probability space, conditional probability, pdf and pmf; discrete and continuous random variables) is quite a challenge.

If you're looking for a more mathematical approach, I can recommend the references in Decomposition of spectrum (functional analysis)#Quantum mechanics. Alternatively, if you want a more easy read, "The Born rule and its interpretation" or the BSc thesis "Rigged Hilbert Space Theory for Hermitian and Quasi-Hermitian Observables" are great as well.

ps. There is actually a very nice video of Feynman explaining the difference between Mathematicians and physicists (YT: "Feynman: Mathematicians versus Physicists") Roffaduft (talk) 04:44, 12 October 2024 (UTC)

No, you don't need to take a Radon-Nikodym derivative or anything. The naïve formula already gives you the probability density in the continuous case. You have to integrate that to get a probability. Tercer (talk) 07:32, 12 October 2024 (UTC)

No, it just doesn't. You are simply wrong about this. Nathaniel Virgo (talk) 10:11, 12 October 2024

edit: sorry, no you're not wrong, I just misinterpreted what you said. You're right that Radon-Nikodym isn't strictly needed, but IMHO it's what justifies using densities in the first place, since it tells you which measures can be expressed that way. Nathaniel Virgo (talk) 10:41, 12 October 2024 (UTC)

A measure is a different thing from a probability density function. The M in POVM stands for measure, not density function. It might be nice and intuitive to think about probability in terms of density functions and not worry about measure theory, and I appreciate that that's what a lot of physicists do a lot of the time, but this article is *about* a measure theoretic concept, and talking about it as if it were a density function is simply not correct. Nathaniel Virgo (talk) 10:14, 12 October 2024 (UTC)

Please don't take the following the wrong way. It would have applied to me at an earlier point in my career. But if your thinking is along the lines of "well of course it has to be a density in the continuous case, what else could it possibly be?", then the issue is probably that you have some misconceptions about how measure theory works. A measure is a function from a sigma algebra to some rig (ring without negatives), which is usually the nonnegative real numbers but in this case it's the rig of positive operators. The elements of the sigma algebra are called events, so a measure assigns probabilities to events (satisfying some axioms). That's literally all it does - the notion of density doesn't appear anywhere in the definition of a measure. One way to come up with a measure is to use a density function, but there are plenty of measures that can't be expressed this way, with delta functions being a classic example. So positive operators valued *measures* are strictly more general than positive operators valued *density functions* would be. And the article most definitely is about measures. Nathaniel Virgo (talk) 10:29, 12 October 2024 (UTC)

This thread is a bit confusing because I initially misinterpreted you. I now agree with what you said initially but I still don't think it makes sense to talk about densities in the article, because the definition given is in terms of measures and not densities.

I think part of the problem is that the article was mixing up three things: a general measure-theoretic definition, a discussion of the discrete case, and the use of the "naive" density based approach. I've edited the section to try and separate the measure theoretic stuff from the discrete case. It would be good if it would also have a discussion of the density approach, but I feel that should go into slightly more detail and give references - if I would try to do it it would turn into "original research", so I'll leave it to someone more familiar with the literature. Nathaniel Virgo (talk) 11:10, 12 October 2024 (UTC)