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Measurement Issues: POVM, Neumann and generalized

  1. Apr 6, 2006 #1

    km

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    [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. jcsd
  3. Apr 7, 2006 #2
    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
     
  4. Apr 8, 2006 #3
    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.
     
  5. Apr 8, 2006 #4
    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
     
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