nLab positive-operator-valued measure



Measure and probability theory


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



Given a measurable space XX and a Hilbert space HH, then a positive-operator–valued measure (POVM) is essentially a probability measure on XX with values in the positive operators on HH.

If all the positive operators appearing are projection operators then one speaks of a projection valued measure (PVM).

Any PVM induces a quantum channel and these are precisely the quantum operations which encode (projective) quantum measurement, see at quantum measurement channel.



Review with extensive references to the literature and historical quotes:

See also:

Discussion in quantum field theory:

Critical comments on the notion of POVMs as “generalized measurements”:

  • Robert B. Griffiths, Section 5 of: Quantum Channels, Kraus Operators, POVMs (2012) [pdf, pdf]

    [p. 17:] When G kG_k has rank greater than one but is not a projector it is not so clear what one learns from the POVM when the outcome is kk. One can assert that in this instance the system a at the time of interest had the property given by the support of G kG_k: the smallest projector P such that PG k=G kP G_k = G_k, see CQT, p. 44. This, however, need not be very informative. In particular there are cases in which the support of G kG_k is I aI_a, in which case the fact that a had this (always true) property tells us nothing.

    Given the generality allowed in the definition of a POVM, this lack of clarity is not too surprising. Think of a situation in which two vehicles collide and a wheel spins off of one of them. What does this tell one about the state of affairs before the collision? Probably it says something, but it might be rather difficult to say just what it is.

    For this reason it is a bit odd to refer to a general POVM as a “measurement.”

Projection valued measures are central objects in the theory of (general) coherent states.

  • Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau, Coherent states, wavelets and their generalizations, Graduate Texts in Contemporary Physics. Springer 2000

Last revised on October 1, 2023 at 09:41:03. See the history of this page for a list of all contributions to it.