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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>
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'''POVM''' 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. 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. 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. Each operator F_i corresponds to a possible measurement outcome.
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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>
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[[File:POV Diagram.png|thumb|280px|Quantum POVM.]]
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[[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.]]
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== Definition ==
== 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
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


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


== Relation to projective measurement ==
== 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
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


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== Naimark's dilation theorem ==
== 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>
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
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
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== Example ==
== 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>
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 ==
== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}


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


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

Latest revision as of 00:31, 24 May 2026

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POVM 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. 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. 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. Each operator F_i corresponds to a possible measurement outcome.

Quantum POVM.

Definition

In the finite-dimensional case, a POVM is a collection of positive semi-definite operators {Fi} acting on a Hilbert space such that

∑i=1nFi=I.

Each operator Fi corresponds to a possible measurement outcome. If the system is in state ρ, then the probability of obtaining outcome i is

Prob⁡(i)=tr⁡(ρFi).

For a pure state |ψ⟩, this becomes

Prob⁡(i)=⟨ψ|Fi|ψ⟩.

Unlike the operators in a projective measurement, the elements of a POVM do not have to be orthogonal projectors.[1][2]

Relation to projective measurement

A projection-valued measure (PVM) is a special case of a POVM in which the measurement operators are orthogonal projections {Πi} satisfying

∑i=1NΠi=I,ΠiΠj=δijΠi.

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.[1]

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.[3] In this sense, POVMs do not replace projective measurements but generalize them by allowing auxiliary systems and larger measurement spaces.[4]

In the finite-dimensional case, if {Fi} is a POVM on a Hilbert space ℋA, then there exists a larger Hilbert space, a projective measurement {Πi}, and an isometry V such that

Fi=V†ΠiV.

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.[1]

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.[5][6]

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ 1.0 1.1 1.2 Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. 
  2. ↑ Davies, Edward Brian (1976). Quantum Theory of Open Systems. London: Academic Press. p. 35. ISBN 978-0-12-206150-9. 
  3. ↑ Gelfand, I. M.; Neumark, M. A. (1943). "On the embedding of normed rings into the ring of operators in Hilbert space". Rec. Math. [Mat. Sbornik] N.S. 12: 197–213. 
  4. ↑ Peres, Asher (1993). Quantum Theory: Concepts and Methods. Kluwer Academic Publishers. 
  5. ↑ J. A. Bergou; U. Herzog; M. Hillery (2004). "Discrimination of Quantum States". Quantum State Estimation. Springer. pp. 417–465. doi:10.1007/978-3-540-44481-7_11. 
  6. ↑ Chefles, Anthony (2000). "Quantum state discrimination". Contemporary Physics 41 (6): 401–424. doi:10.1080/00107510010002599. 


Author: Harold Foppele


Source attribution: Physics:Quantum POVM

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