# Measurement Issues: POVM, Neumann and generalized

1. Apr 6, 2006

### km

[SOLVED] Measurement Issues: POVM, Neumann and generalized

What is a POVM measurement? How is it different from Nuemann
measurement theory and what would be the most general measurement
formulation?
Please keep things discrete and not continuous.
-KM

2. Apr 7, 2006

### Arnold Neumaier

km wrote:
> What is a POVM measurement? How is it different from Nuemann
> measurement theory

It is much more realistic - von Neumann measurements are highly
idealized. For example, the latter don't capture such things as
approximate joint measurements of position and momentum (to
accuracies allowed by the uncertainty relation).

> and what would be the most general measurement formulation?

The POVM is the most general formulation in current use.

http://web.mit.edu/kayla/Public/preskel_notes/book2.pdf
is a set of lecture notes about quantum computation
which gives a reasonable background on measurement models.

p. 39 features the ideal Copenhagen measurement of a pure
quantum system; both pureness and ideal measurement are,
however, unrealistic idealizations in most circumstances.

p. 49ff discusses the density matrix, describing more realistic
quantum systems, and p. 81ff discusses POVMs, describing more
realistic measurements.

Arnold Neumaier

Last edited by a moderator: Apr 22, 2017
3. Apr 8, 2006

### Vonny N.

Here's a gentle summary:

In the orthodox formulation of quantum mechanics, real observables are
represented by self-adjoint operators. The thing that is really special
about these operators is that they can be represented as
Projection-Valued Measures (PVMs). In the three-dimensional case this
means that each possible value of the observable is 'linked to' an
orthogonal projection operator in such a way that the self adjoint
operator can be written: A = aP + bQ + cR, where P, Q, and R are
mutually orthogonal and sum to the identity (i.e. P+Q+R=I). This turns
out to be equivalent to saying that A can be written in an appropriate
basis as diag(a,b,c), but the PVM form turns out to lead to an
important generalization - namely the Positive Operator-Valued Measure
(POVM).

There are lots of arguments for the utility of the POVM but one
important one is that they allow us to extend the standard formalism in
a consistent way to allow us to admit 'fuzzy' measurements rather than
perfectly 'clean' ones. The extension consists in demanding only that
the P, Q, and R be postive Hermitian operators which sum to the
identity, which entails the PVM as a special case. In the case of a
POVM the value of the expression <a|P|a> could be interpreted, for
example, as the probability of outcome 'a' occurring in a measuring
arrangement which sometimes makes mistakes about the thing it is
'trying' to measure.

One reason they are trickier to work with is that they don't in general
have a unique representation as a self-adjoint operator, which means
you have to work with measure-theoretic constructs rather than the more
ubiquitous algebra of our old friend Dirac. But since there are
important problems that can only be solved using POVMs, it looks like
they're here to stay.

Hope this helps a little.

Vonny N.

4. Apr 8, 2006

### Oh No

Thus spake km <mailkaran@gmail.com>
>What is a POVM measurement? How is it different from Nuemann
>measurement theory and what would be the most general measurement
>formulation?
>Please keep things discrete and not continuous.
>-KM
>

I don't think I can do better than reference you to wikipedia

http://en.wikipedia.org/wiki/POVM

The orthogonal projectors referred to there are the Von Neumann
measurement operators. It explains the distinction quite nicely.

Regards

--
Charles Francis
substitute charles for NotI to email