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 MathematicsPrevious 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

Theory of Probability and Its ApplicationsMar 1, 197991 citations

Violation of general Bell inequalities by a pure bipartite quantum state

Loubenets, Elena R., Min Namkung

In the present article, based on the formalism introduced in [Loubenets, J. Math. Phys. 53, 022201 (2012)], we derive for a pure bipartite quantum state a new upper bound on its maximal violation of general Bell inequalities. This new bound indicates that, for an infinite dimensional pure bipartite state with a finite sum of its Schmidt coefficients, violation of any general Bell inequality is bounded from above by the value independent on a number of settings and a type of outcomes, continuous or discrete, specific to this Bell inequality. As an example, we apply our new general results to specifying upper bounds on the maximal violation of general Bell inequalities by infinite dimensional bipartite states having the Bell states like forms comprised of two binary coherent states $|α\rangle ,|-α\rangle$ , with $α>0$ . We show that, for each of these bipartite coherent states, the maximal violation of general Bell inequalities cannot exceed the value $3$ and analyse numerically the dependence of the derived analytical upper bounds on a parameter $α>0$ .In the present article, based on the formalism introduced in [Loubenets, J. Math. Phys. 53, 022201 (2012)], we derive for a pure bipartite quantum state a new upper bound on its maximal violation of general Bell inequalities. This new bound indicates that, for an infinite dimensional pure bipartite state with a finite sum of its Schmidt coefficients, violation of any general Bell inequality is bounded from above by the value independent on a number of settings and a type of outcomes, continuous or discrete, specific to this Bell inequality. As an example, we apply our new general results to specifying upper bounds on the maximal violation of general Bell inequalities by infinite dimensional bipartite states having the Bell states like forms comprised of two binary coherent states $|α\rangle ,|-α\rangle$, with $α>0$. We show that, for each of these bipartite coherent states, the maximal violation of general Bell inequalities cannot exceed the value $3$ and analyse numerically the dependence of the derived analytical upper bounds on a parameter $α>0$.

arXiv:2110.08859Oct 17, 20210 citations

Some results and problems on quantum Bell-type inequalities

Boris S. Tsirelson

Jan 1, 1993198 citations

Inequalities for Purity Parameters of Multiqudit and Single-Qudit States

Vladimir I. Man’ko, Liubov A. Markovich

Journal of Russian Laser ResearchMar 1, 20161 citation

|α\rangle ,|-α\rangle

, with

α>0

. We show that, for each of these bipartite coherent states, the maximal violation of general Bell inequalities cannot exceed the value

3

and analyse numerically the dependence of the derived analytical upper bounds on a parameter

α>0