Talk:Projection-valued measure

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Standard Borel space

Is a standard Borel space the same as a Borel space? -- 10:51, 28 October 2005 Oleg Alexandrov

Standard Borel space: Borel structure of a Polish space. -- 19:12, 28 October 2005 CSTAR

Great! Can I challenge you to fill in the redlink? Oleg Alexandrov (talk) 09:57, 29 October 2005 (UTC)[reply]

Vector measure

Re: recent addition:

It is further generalized by vector measures which take values in any Banach space.

This is actually quite subtle. By definition, a projection-valued measure P is σ-additive relative to the weak (or strong) operator topology on the set of self-adjoint projections. This means the P measure of a monotone union of a sequence of sets is the operator weak limit of the P measures of the elements of the sequence. This is equivalent to P being σ-additive relative to the ultraweak topology, since the ultrweak and weak operator toplogies coincide on bounded sets. The ultraweak topology in turn is a weak* topology (the algebra L(H) of bounded operators on H is the dual of the Banach space of trace class operators). However, weak* topologies require additional structure to define, i.e. being a dual space. Note that the weak* topology on a dual Banach space E* is weaker in general than the weak topology on E* considered as a Banach space in its own right.

So one should actually say that a projection-valued measure is generalized by a vector measure which which takes values in a dual Banach space and is weak* continuous.

I hope I haven't said anything silly here. --CSTAR 17:03, 9 February 2007 (UTC)[reply]

I have a weak* headache after reading all that. But thanks, I got your point. Oleg Alexandrov (talk) 02:44, 10 February 2007 (UTC)[reply]

Math vs. physics

This article is written for a mathematician, and is fairly nice and clear. The POVM is written for physicists, and is a confusing mess. Worse, they seem to have very different poiints of view: here, we have sigma algebras fibered with hilbert spaces. Over there, the sigma-algebra is very nearly absent, making a tiny appearence. Not a single hint of anything being fibered, over there. Its very hard to come to the conclusion that these two toipics have anything to do with one-another. It would be great if this was fixed (viz, add a plain-physics intro to this article, and remove some of the gibberish from the other article). User:Linas (talk) 22:44, 23 November 2013 (UTC)[reply]

I agree 100%.

I added a section relating the definitions to quantum mechanics. Hopefully I did not write too much; I know that the same definitions appear in many places, but I find it useful to repeat them in the present article in order to cement the connection with this particular notation.

I took the liberty of defining the term "projective measurement" as a measurement that can be effectuated by a projection-valued measurement, even though I have not seen this defined. It is used in a number of articles in Wikipedia and I assume from context that this is the meaning. Is it a recent term due to quantum information theory, or is it an older term from the theory of quantum mechanics? The concept is old of course, but maybe the perspective is new.

89.217.0.87 (talk) 14:34, 3 May 2015 (UTC)[reply]

Orthogonality condition in the definition

Is the condition "If π is a projection-valued measure and E \cap F = \emptyset, then π(E), π(F) are orthogonal projections" part of the definition of a projection-valued measure? The current formulation of the definition seems to suggest that this orthogonality condition follows from the lines above it. I don't see how this is possible since we can consider π(E)=m(E)*id_H where m is any probability measure on (X,M).