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Simplified and robust conditions for optimum testing of multiple hypotheses in quantum detection theory

Jon E. Tyson

Published Feb 3, 2009• 0 citations
OpenAlex

Abstract

We construct crude estimates for non-optimality of quantum measurements in terms of their violation of Holevo's simplified minimum-error optimality conditions. As an application, we show that a modification of Barnett and Croke's proof of the optimality conditions yields a convergent iterative scheme for computing optimal measurements.

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Estimates of non-optimality of quantum measurements and a simple iterative method for computing optimal measurements

Tyson, Jon

We construct crude estimates for non-optimality of quantum measurements in terms of their violation of Holevo's simplified minimum-error optimality conditions. As an application, we show that a modification of Barnett and Croke's proof of the optimality conditions yields a convergent iterative scheme for computing optimal measurements.

arXiv:0902.0395•Feb 3, 2009•3 citations

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The problem of specifying the optimum quantum detector in multiple hypotheses testing is considered for application to optical communications. The quantum digital detection problem is formulated as a linear programming problem on an infinite-dimensional space. A necessary and sufficient condition is derived by the application of a general duality theorem specifying the optimum detector in terms of a set of linear operator equations and inequalities. Existence of the optimum quantum detector is also established. The optimality of commuting detection operators is discussed in some examples. The structure and performance of the optimal receiver are derived for the quantum detection of narrow-band coherent orthogonal and simplex signals. It is shown that modal photon counting is asymptotically optimum in the limit of a large signaling alphabet and that the capacity goes to infinity in the absence of a bandwidth limitation.

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This lecture notes gives a basic survey of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.

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Quantum hypothesis testing and state discrimination

J. Prabhu Tej, Syed Raunaq Ahmed, A. R. Usha Devi +1 more

This expository article gives an overview of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.

arXiv:1803.04944•Mar 13, 2018•5 citations

On Asymptotically Optimal Hypothesis Testing in Quantum Statistics

Александр Семенович Холево

Previous article Next article On Asymptotically Optimal Hypothesis Testing in Quantum StatisticsA. S. KholevoA. S. Kholevohttps://doi.org/10.1137/1123048PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. S. Holevo, Statistical decision theory for quantum systems, J. Multivariate Anal., 3 (1973), 337–394 10.1016/0047-259X(73)90028-6 MR0337245 0275.62004 CrossrefGoogle Scholar[2] C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, N. Y., 1976 Google Scholar[3] Horace P. Yuen, , Robert S. Kennedy and , Melvin Lax, Optimum testing of multiple hypotheses in quantum detection theory, IEEE Trans. Information Theory, IT-21 (1975), 125–134 10.1109/TIT.1975.1055351 MR0397511 0301.94001 CrossrefGoogle Scholar[4] A. S. Kholevo, Remarks on optimal quantum measurements, Problems in Transmission of Information, 10 (1974), 51–55, (In Russian.) 0309.94001 Google Scholar[5] Robert Schatten, Norm ideals of completely continuous operators, Second printing. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 27, Springer-Verlag, Berlin, 1970vii+81 MR0257800 0188.44103 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Quantum Algorithm for Petz Recovery Channels and Pretty Good MeasurementsPhysical Review Letters, Vol. 128, No. 22 | 1 June 2022 Cross Ref Fast algorithm for quantum polar decomposition and applicationsPhysical Review Research, Vol. 4, No. 1 | 22 February 2022 Cross Ref Reversing Lindblad Dynamics via Continuous Petz Recovery MapPhysical Review Letters, Vol. 128, No. 2 | 11 January 2022 Cross Ref Symmetric distinguishability as a quantum resourceNew Journal of Physics, Vol. 23, No. 8 | 9 August 2021 Cross Ref Quantum Keyless Private Communication Versus Quantum Key Distribution for Space LinksPhysical Review Applied, Vol. 16, No. 1 | 2 July 2021 Cross Ref Quantum Ranging with Gaussian EntanglementPhysical Review Letters, Vol. 126, No. 24 | 14 June 2021 Cross Ref Simple Upper and 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Ref REVERSIBILITY CONDITIONS FOR QUANTUM OPERATIONSReviews in Mathematical Physics, Vol. 24, No. 07 | 1 Aug 2012 Cross Ref Quantum tomography via unambiguous state discriminationPhysical Review A, Vol. 86, No. 1 | 1 July 2012 Cross Ref Achieving the Holevo bound via sequential measurementsPhysical Review A, Vol. 85, No. 1 | 1 January 2012 Cross Ref How Many Copies are Needed for State Discrimination?IEEE Transactions on Information Theory, Vol. 58, No. 1 | 1 Jan 2012 Cross Ref Minimum error discrimination between similarity-transformed quantum statesPhysical Review A, Vol. 84, No. 1 | 1 July 2011 Cross Ref Sequential Projective Measurements for Channel DecodingPhysical Review Letters, Vol. 106, No. 25 | 1 June 2011 Cross Ref Min-entropy and quantum key distribution: Nonzero key rates for "small" numbers of signalsPhysical Review A, Vol. 83, No. 2 | 1 February 2011 Cross Ref Asymptotically Optimal Discrimination between Pure Quantum StatesTheory of Quantum Computation, Communication, and Cryptography | 1 Jan 2011 Cross Ref Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iteratesJournal of Mathematical Physics, Vol. 51, No. 9 | 1 Sep 2010 Cross Ref Exponential error rates in multiple state discrimination on a quantum spin chainJournal of Mathematical Physics, Vol. 51, No. 7 | 1 Jul 2010 Cross Ref Qubit state discriminationPhysical Review A, Vol. 81, No. 6 | 1 June 2010 Cross Ref Discrimination of quantum statesJournal of Modern Optics, Vol. 57, No. 3 | 10 Feb 2010 Cross Ref Optimum unambiguous discrimination of linearly independent pure statesPhysical Review A, Vol. 80, No. 5 | 1 November 2009 Cross Ref Unambiguous modification of nonorthogonal single- and two-photon polarization statesPhysical Review A, Vol. 79, No. 5 | 15 May 2009 Cross Ref Quantum state discriminationAdvances in Optics and Photonics, Vol. 1, No. 2 | 11 February 2009 Cross Ref Error rates of Belavkin weighted 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Quantum DetectionCommunications, Information and Network Security | 1 Jan 2003 Cross Ref Optimal tight frames and quantum measurementIEEE Transactions on Information Theory, Vol. 48, No. 3 | 1 Mar 2002 Cross Ref Reversing quantum dynamics with near-optimal quantum and classical fidelityJournal of Mathematical Physics, Vol. 43, No. 5 | 1 Jan 2002 Cross Ref Optimal phase estimation and square root measurementJournal of Physics A: Mathematical and General, Vol. 34, No. 35 | 24 August 2001 Cross Ref Fidelity and the communication of quantum informationJournal of Physics A: Mathematical and General, Vol. 34, No. 35 | 24 August 2001 Cross Ref Minimum-error discrimination between multiply symmetric statesPhysical Review A, Vol. 64, No. 3 | 1 August 2001 Cross Ref Compression of quantum-measurement operationsPhysical Review A, Vol. 64, No. 1 | 1 June 2001 Cross Ref Quantum channels showing superadditivity in classical capacityPhysical Review A, Vol. 58, No. 1 | 1 July 1998 Cross Ref A demonstration of superadditivity in the classical capacity of a quantum channelPhysics Letters A, Vol. 236, No. 1-2 | 1 Dec 1997 Cross Ref On the design of nonlinear discrete-time predictors (Corresp.)IEEE Transactions on Information Theory, Vol. 28, No. 2 | 1 Mar 1982 Cross Ref Bayes-cost reduction algorithm in quantum hypothesis testing (Corresp.)IEEE Transactions on Information Theory, Vol. 28, No. 2 | 1 Mar 1982 Cross Ref Volume 23, Issue 2| 1979Theory of Probability & Its Applications227-460 History Submitted:22 March 1977Published online:17 July 2006 InformationCopyright © 1979 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1123048Article page range:pp. 411-415ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Theory of Probability and Its Applications•Mar 1, 1979•91 citations

Bayes-cost reduction algorithm in quantum hypothesis testing (Corresp.)

Carl W. Helstrom

An iterative procedure is described for reducing the Bayes cost in decisions amongM>2quantum hypotheses by minimizing the average cost in binary decisions between all possible pairs of hypotheses: the resulting decision strategy is a projection-valued measure and yields an upper bound to the minimum attainable Bayes cost. From it is derived an algorithm for finding the optimum measurement states for choosing amongMlinearly independent pure states with minimum probability of error. The method is also applied to decisions amongMunimodal coherent quantum signals in thermal noise.

IEEE Transactions on Information Theory•Mar 1, 1982•56 citations
We construct crude estimates for non-optimality of quantum measurements in terms of their violation of Holevo's simplified minimum-error optimality conditions. As an application, we show that a modification of Barnett and Croke's proof of the optimality conditions yields a convergent iterative scheme for computing optimal measurements.
The problem of specifying the optimum quantum detector in multiple hypotheses testing is considered for application to optical communications. The quantum digital detection problem is formulated as a linear programming problem on an infinite-dimensional space. A necessary and sufficient condition is derived by the application of a general duality theorem specifying the optimum detector in terms of a set of linear operator equations and inequalities. Existence of the optimum quantum detector is also established. The optimality of commuting detection operators is discussed in some examples. The structure and performance of the optimal receiver are derived for the quantum detection of narrow-band coherent orthogonal and simplex signals. It is shown that modal photon counting is asymptotically optimum in the limit of a large signaling alphabet and that the capacity goes to infinity in the absence of a bandwidth limitation.
This lecture notes gives a basic survey of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.
This expository article gives an overview of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.
Previous article Next article On Asymptotically Optimal Hypothesis Testing in Quantum StatisticsA. S. KholevoA. S. Kholevohttps://doi.org/10.1137/1123048PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. S. Holevo, Statistical decision theory for quantum systems, J. Multivariate Anal., 3 (1973), 337–394 10.1016/0047-259X(73)90028-6 MR0337245 0275.62004 CrossrefGoogle Scholar[2] C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, N. Y., 1976 Google Scholar[3] Horace P. Yuen, , Robert S. Kennedy and , Melvin Lax, Optimum testing of multiple hypotheses in quantum detection theory, IEEE Trans. Information Theory, IT-21 (1975), 125–134 10.1109/TIT.1975.1055351 MR0397511 0301.94001 CrossrefGoogle Scholar[4] A. S. Kholevo, Remarks on optimal quantum measurements, Problems in Transmission of Information, 10 (1974), 51–55, (In Russian.) 0309.94001 Google Scholar[5] Robert Schatten, Norm ideals of completely continuous operators, Second printing. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 27, Springer-Verlag, Berlin, 1970vii+81 MR0257800 0188.44103 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Quantum Algorithm for Petz Recovery Channels and Pretty Good MeasurementsPhysical Review Letters, Vol. 128, No. 22 | 1 June 2022 Cross Ref Fast algorithm for quantum polar decomposition and applicationsPhysical Review Research, Vol. 4, No. 1 | 22 February 2022 Cross Ref Reversing Lindblad Dynamics via Continuous Petz Recovery MapPhysical Review Letters, Vol. 128, No. 2 | 11 January 2022 Cross Ref Symmetric distinguishability as a quantum resourceNew Journal of Physics, Vol. 23, No. 8 | 9 August 2021 Cross Ref Quantum Keyless Private Communication Versus Quantum Key Distribution for Space LinksPhysical Review Applied, Vol. 16, No. 1 | 2 July 2021 Cross Ref Quantum Ranging with Gaussian EntanglementPhysical Review Letters, Vol. 126, No. 24 | 14 June 2021 Cross Ref Simple Upper and Lower Bounds on the Ultimate Success Probability for Discriminating Arbitrary Finite-Dimensional Quantum ProcessesPhysical Review Letters, Vol. 126, No. 20 | 18 May 2021 Cross Ref Idler-free channel position findingPhysical Review A, Vol. 103, No. 4 | 29 April 2021 Cross Ref Quantum state discrimination on reconfigurable noise-robust quantum networksPhysical Review Research, Vol. 2, No. 4 | 2 October 2020 Cross Ref Ultimate Limits for Multiple Quantum Channel DiscriminationPhysical Review Letters, Vol. 125, No. 8 | 21 August 2020 Cross Ref Maximal trace distance between isoenergetic bosonic Gaussian statesJournal of Mathematical Physics, Vol. 58, No. 12 | 1 Dec 2017 Cross Ref Nonlocality free wirings and the distinguishability between Bell boxesPhysical Review A, Vol. 95, No. 3 | 17 March 2017 Cross Ref Geometric Measures of Quantum Correlations with Bures and Hellinger DistancesLectures on General Quantum Correlations and their Applications | 25 June 2017 Cross Ref Peculiarities of qubit initial-state preparation by nonselective measurements on an overcomplete basisPhysical Review A, Vol. 92, No. 6 | 9 December 2015 Cross Ref Entropic Lower Bound for Distinguishability of Quantum StatesAdvances in Mathematical Physics, Vol. 2015 | 1 Jan 2015 Cross Ref Iterative methods for finding optimal quantum measurements under minimum-error and minimax criteriaPhysical Review A, Vol. 91, No. 1 | 14 January 2015 Cross Ref Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discriminationJournal of Mathematical Physics, Vol. 55, No. 10 | 1 Oct 2014 Cross Ref Quantum correlations and distinguishability of quantum statesJournal of Mathematical Physics, Vol. 55, No. 7 | 1 Jul 2014 Cross Ref Quantum reading of digital memory with non-Gaussian entangled lightPhysical Review A, Vol. 87, No. 5 | 1 May 2013 Cross Ref Quantum state discrimination bounds for finite sample sizeJournal of Mathematical Physics, Vol. 53, No. 12 | 1 Dec 2012 Cross Ref REVERSIBILITY CONDITIONS FOR QUANTUM OPERATIONSReviews in Mathematical Physics, Vol. 24, No. 07 | 1 Aug 2012 Cross Ref Quantum tomography via unambiguous state discriminationPhysical Review A, Vol. 86, No. 1 | 1 July 2012 Cross Ref Achieving the Holevo bound via sequential measurementsPhysical Review A, Vol. 85, No. 1 | 1 January 2012 Cross Ref How Many Copies are Needed for State Discrimination?IEEE Transactions on Information Theory, Vol. 58, No. 1 | 1 Jan 2012 Cross Ref Minimum error discrimination between similarity-transformed quantum statesPhysical Review A, Vol. 84, No. 1 | 1 July 2011 Cross Ref Sequential Projective Measurements for Channel DecodingPhysical Review Letters, Vol. 106, No. 25 | 1 June 2011 Cross Ref Min-entropy and quantum key distribution: Nonzero key rates for "small" numbers of signalsPhysical Review A, Vol. 83, No. 2 | 1 February 2011 Cross Ref Asymptotically Optimal Discrimination between Pure Quantum StatesTheory of Quantum Computation, Communication, and Cryptography | 1 Jan 2011 Cross Ref Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iteratesJournal of Mathematical Physics, Vol. 51, No. 9 | 1 Sep 2010 Cross Ref Exponential error rates in multiple state discrimination on a quantum spin chainJournal of Mathematical Physics, Vol. 51, No. 7 | 1 Jul 2010 Cross Ref Qubit state discriminationPhysical Review A, Vol. 81, No. 6 | 1 June 2010 Cross Ref Discrimination of quantum statesJournal of Modern Optics, Vol. 57, No. 3 | 10 Feb 2010 Cross Ref Optimum unambiguous discrimination of linearly independent pure statesPhysical Review A, Vol. 80, No. 5 | 1 November 2009 Cross Ref Unambiguous modification of nonorthogonal single- and two-photon polarization statesPhysical Review A, Vol. 79, No. 5 | 15 May 2009 Cross Ref Quantum state discriminationAdvances in Optics and Photonics, Vol. 1, No. 2 | 11 February 2009 Cross Ref Error rates of Belavkin weighted quantum measurements and a converse to Holevo's asymptotic optimality theoremPhysical Review A, Vol. 79, No. 3 | 31 March 2009 Cross Ref Two-sided estimates of minimum-error distinguishability of mixed quantum states via generalized Holevo–Curlander boundsJournal of Mathematical Physics, Vol. 50, No. 3 | 1 Mar 2009 Cross Ref Asymptotic Error Rates in Quantum Hypothesis TestingCommunications in Mathematical Physics, Vol. 279, No. 1 | 8 February 2008 Cross Ref Quantum Chernoff bound as a measure of distinguishability between density matrices: Application to qubit and Gaussian statesPhysical Review A, Vol. 77, No. 3 | 1 March 2008 Cross Ref Quantum state discrimination and selected applicationsJournal of Physics: Conference Series, Vol. 84 | 12 October 2007 Cross Ref Discriminating States: The Quantum Chernoff BoundPhysical Review Letters, Vol. 98, No. 16 | 1 April 2007 Cross Ref Tomographic quantum cryptographyPhysical Review A, Vol. 68, No. 2 | 1 August 2003 Cross Ref Advances in Quantum DetectionCommunications, Information and Network Security | 1 Jan 2003 Cross Ref Optimal tight frames and quantum measurementIEEE Transactions on Information Theory, Vol. 48, No. 3 | 1 Mar 2002 Cross Ref Reversing quantum dynamics with near-optimal quantum and classical fidelityJournal of Mathematical Physics, Vol. 43, No. 5 | 1 Jan 2002 Cross Ref Optimal phase estimation and square root measurementJournal of Physics A: Mathematical and General, Vol. 34, No. 35 | 24 August 2001 Cross Ref Fidelity and the communication of quantum informationJournal of Physics A: Mathematical and General, Vol. 34, No. 35 | 24 August 2001 Cross Ref Minimum-error discrimination between multiply symmetric statesPhysical Review A, Vol. 64, No. 3 | 1 August 2001 Cross Ref Compression of quantum-measurement operationsPhysical Review A, Vol. 64, No. 1 | 1 June 2001 Cross Ref Quantum channels showing superadditivity in classical capacityPhysical Review A, Vol. 58, No. 1 | 1 July 1998 Cross Ref A demonstration of superadditivity in the classical capacity of a quantum channelPhysics Letters A, Vol. 236, No. 1-2 | 1 Dec 1997 Cross Ref On the design of nonlinear discrete-time predictors (Corresp.)IEEE Transactions on Information Theory, Vol. 28, No. 2 | 1 Mar 1982 Cross Ref Bayes-cost reduction algorithm in quantum hypothesis testing (Corresp.)IEEE Transactions on Information Theory, Vol. 28, No. 2 | 1 Mar 1982 Cross Ref Volume 23, Issue 2| 1979Theory of Probability & Its Applications227-460 History Submitted:22 March 1977Published online:17 July 2006 InformationCopyright © 1979 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1123048Article page range:pp. 411-415ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics
An iterative procedure is described for reducing the Bayes cost in decisions amongM>2quantum hypotheses by minimizing the average cost in binary decisions between all possible pairs of hypotheses: the resulting decision strategy is a projection-valued measure and yields an upper bound to the minimum attainable Bayes cost. From it is derived an algorithm for finding the optimum measurement states for choosing amongMlinearly independent pure states with minimum probability of error. The method is also applied to decisions amongMunimodal coherent quantum signals in thermal noise.