nLab positive-operator-valued measure

Idea

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

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.

References

Monographs:

• Paul Busch, Marian Grabowski, Pekka J. Lahti, Operational Quantum Physics, Lecture Notes in Physics Monographs 31, Springer (1995) [doi:10.1007/978-3-540-49239-9]

• Paul Busch, Pekka J. Lahti, Juha-Pekka Pellonpää, Kari Ylinen, Quantum Measurement, Springer (2016) [doi:10.1007/978-3-319-43389-9]

Review with extensive references to the literature and historical quotes:

• Arnold Neumaier, Born’s rule and measurement [arXiv:1912.09906]

See also:

• John Preskill, Generalized measurements, Section 3.1.2 [pdf] in: Quantum Computation, lecture notes, since 2004 [web]

• Stephen Barnett, Chapter 4 of: Quantum Information, Oxford University Press (2009) [ISBN:9780198527633]

• Mario Flory, POVMs and superoperators (2011) [pdf, pdf]

• Nicholas Wheeler, Generalized Quantum Measurement (2012) [pdf, pdf]

• Wikipedia, POVM

Discussion in quantum field theory:

• Adam Bednorz, General quantum measurements in relativistic quantum field theory, Phys. Rev. D 108 056020 (2023) [arXiv:2305.12765, doi:10.1103/PhysRevD.108.056020]

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 $G_k$ 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 $k$. One can assert that in this instance the system a at the time of interest had the property given by the support of $G_k$: the smallest projector P such that $P G_k = G_k$, see CQT, p. 44. This, however, need not be very informative. In particular there are cases in which the support of $G_k$ is $I_a$, 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