Physics:Quantum POVM - Revision history

WikiHarold: Restore missing Quantum reference definitions

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

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

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	'''POVM''' is a ~~Book I topic in the Quantum Collection. 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.		'''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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	== 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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	== 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 ==

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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>		'''POVM''' is a Book I topic in the Quantum Collection. 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.

	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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	== 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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	== 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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			{{Short description\|Quantum Collection topic on Quantum POVM}}

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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>		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>		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.]]~~		</div>

	== Definition ==		== Definition ==

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