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{{Quantum book backlink|Measurement and information}}

A '''positive operator-valued measure''' ('''POVM''') is a generalized mathematical description of measurement in [[quantum mechanics]]. POVMs extend the notion of [[projective measurement]] by allowing measurement outcomes to be associated with positive operators that need not be orthogonal projections.<ref name=Nielsen>{{cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2000}}</ref><ref name=Davies>{{cite book |last=Davies |first=Edward Brian |title=Quantum Theory of Open Systems |publisher=Academic Press |location=London |year=1976 |page=35 |isbn=978-0-12-206150-9}}</ref>

POVMs are the most general kind of measurement commonly used in quantum theory and are especially important in [[quantum information science]], where they describe realistic or optimized measurements that cannot always be represented by projection-valued measures alone.<ref name=PeresTerno>{{cite journal |last1=Peres |first1=Asher |last2=Terno |first2=Daniel R. |title=Quantum information and relativity theory |journal=Reviews of Modern Physics |volume=76 |issue=1 |year=2004 |pages=93–123 |doi=10.1103/RevModPhys.76.93}}</ref>

[[File:POV Diagram.png|thumb|400px|right|A POVM can distinguish non-orthogonal quantum states more effectively than a projective measurement in tasks such as unambiguous state discrimination.]]

== Definition ==
In the finite-dimensional case, a POVM is a collection of positive semi-definite operators <math>\{F_i\}</math> acting on a [[Hilbert space]] such that

<math display="block">
\sum_{i=1}^n F_i = I.
</math>

Each operator <math>F_i</math> corresponds to a possible measurement outcome. If the system is in state <math>\rho</math>, then the probability of obtaining outcome <math>i</math> is

<math display="block">
\operatorname{Prob}(i) = \operatorname{tr}(\rho F_i).
</math>

For a pure state <math>|\psi\rangle</math>, this becomes

<math display="block">
\operatorname{Prob}(i) = \langle \psi | F_i | \psi \rangle.
</math>

Unlike the operators in a projective measurement, the elements of a POVM do not have to be orthogonal projectors.<ref name=Nielsen/><ref name=Davies/>

== Relation to projective measurement ==
A [[projection-valued measure]] (PVM) is a special case of a POVM in which the measurement operators are orthogonal projections <math>\{\Pi_i\}</math> satisfying

<math display="block">
\sum_{i=1}^N \Pi_i = I, \quad \Pi_i \Pi_j = \delta_{ij}\Pi_i.
</math>

Thus every projective measurement defines a POVM, but not every POVM is projective. Because POVM elements need not be orthogonal, a POVM can have more outcomes than the dimension of the Hilbert space and can describe measurements that are impossible in the projective framework alone.<ref name=Nielsen/>

== Naimark's dilation theorem ==
A fundamental result, [[Naimark's dilation theorem]], shows that every POVM can be realized as a projective measurement on a larger Hilbert space.<ref name=Naimark>{{cite journal |last1=Gelfand |first1=I. M. |last2=Neumark |first2=M. A. |title=On the embedding of normed rings into the ring of operators in Hilbert space |journal=Rec. Math. [Mat. Sbornik] N.S. |volume=12 |year=1943 |pages=197–213}}</ref> In this sense, POVMs do not replace projective measurements but generalize them by allowing auxiliary systems and larger measurement spaces.<ref name=PeresBook>{{cite book |last=Peres |first=Asher |title=Quantum Theory: Concepts and Methods |publisher=Kluwer Academic Publishers |year=1993}}</ref>

In the finite-dimensional case, if <math>\{F_i\}</math> is a POVM on a Hilbert space <math>\mathcal{H}_A</math>, then there exists a larger Hilbert space, a projective measurement <math>\{\Pi_i\}</math>, and an isometry <math>V</math> such that

<math display="block">
F_i = V^\dagger \Pi_i V.
</math>

This provides the standard physical interpretation of generalized measurements.

== Post-measurement state ==
A POVM determines the probabilities of outcomes, but by itself it does not uniquely determine the post-measurement quantum state. Different physical implementations can realize the same POVM while producing different state changes.<ref name=Nielsen/>

To specify both the outcome probabilities and the resulting state, one uses the more detailed concept of a '''quantum instrument'''. Thus a POVM describes the statistical part of a measurement, while an instrument describes the full measurement process.

== Example ==
An important example is '''unambiguous quantum state discrimination''', in which one wants to distinguish two non-orthogonal states without ever making an incorrect identification, at the cost of sometimes obtaining an inconclusive result. POVMs can accomplish this more efficiently than projective measurements and therefore play a central role in [[quantum information]] and [[quantum cryptography]].<ref name=Bergou>{{cite book |author1=J. A. Bergou |author2=U. Herzog |author3=M. Hillery |editor1=M. Paris |editor2=J. Řeháček |title=Quantum State Estimation |publisher=Springer |year=2004 |pages=417–465 |chapter=Discrimination of Quantum States |doi=10.1007/978-3-540-44481-7_11}}</ref><ref name=Chefles>{{cite journal |last=Chefles |first=Anthony |title=Quantum state discrimination |journal=Contemporary Physics |volume=41 |issue=6 |year=2000 |pages=401–424 |doi=10.1080/00107510010002599}}</ref>

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum POVM|1}}

Physics:Quantum POVM - Revision history

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