Talk:POVM

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[edit] 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)

[edit] 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)