Talk:POVM

message: 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)
thread_title:  Analogy

message: 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)
replies: text: They are the same, by the polarization identity of the inner product. Mct mht (talk) 12:24, 26 November 2013 (UTC)
replies: text: I added a remark that makes this clear. 89.217.0.87 (talk) 14:49, 3 May 2015 (UTC)
thread_title:  real vs. complex 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: I agree, the article needs a lot of work. 89.217.0.87 (talk) 14:41, 3 May 2015 (UTC)
thread_title: Schizophrenic content

message: 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)
thread_title:  Possible non-consensus claim 

message: 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)
thread_title:  The assumptions for the very interesting example need more work 

message: 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)
thread_title: Polarizers and number of modes at the end of the very interesting example