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Quantum chi-squared and goodness of fit testing

Temme, Kristan and Verstraete, Frank (2015) Quantum chi-squared and goodness of fit testing. Journal of Mathematical Physics, 56 (1). Art. No. 012202. ISSN 0022-2488. http://resolver.caltech.edu/CaltechAUTHORS:20150306-101642390

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Abstract

A quantum mechanical hypothesis test is presented for the hypothesis that a certain setup produces a given quantum state. Although the classical and the quantum problems are very much related to each other, the quantum problem is much richer due to the additional optimization over the measurement basis. A goodness of fit test for i.i.d quantum states is developed and a max-min characterization for the optimal measurement is introduced. We find the quantum measurement which leads both to the maximal Pitman and Bahadur efficiencies, and determine the associated divergence rates. We discuss the relationship of the quantum goodness of fit test to the problem of estimating multiple parameters from a density matrix. These problems are found to be closely related and we show that the largest error of an optimal strategy, determined by the smallest eigenvalue of the Fisher information matrix, is given by the divergence rate of the goodness of fit test.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1063/1.4905843 DOIArticle
http://scitation.aip.org/content/aip/journal/jmp/56/1/10.1063/1.4905843PublisherArticle
http://arxiv.org/abs/1112.6343arXivDiscussion Paper
Additional Information:© 2015 AIP Publishing LLC. Received 29 August 2014; accepted 31 December 2014; published online 22 January 2015. We would like to thank Koenraad Audenaert for the help in formulating the proof of Lemma 5. This work was supported by the EU Strep project QUEVADIS, the ERC grant QUERG, and the FWF SFB grants FoQuS and ViCoM. K.T. is grateful for the support from the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation (Grant Nos. PHY-0803371 and PHY-1125565). An important part of this work was done at Stony Brook.
Group:IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
EU Strep Project QUEVADISUNSPECIFIED
ERC Grant QUERGUNSPECIFIED
FWF SFB Grant FoQuSUNSPECIFIED
FWF SFB Grant ViCoMUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
NSF Physics Frontiers CenterUNSPECIFIED
Gordon and Betty Moore FoundationUNSPECIFIED
NSFPHY-0803371
NSFPHY-1125565
Record Number:CaltechAUTHORS:20150306-101642390
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20150306-101642390
Official Citation:Quantum chi-squared and goodness of fit testing Temme, Kristan and Verstraete, Frank, Journal of Mathematical Physics, 56, 012202 (2015), DOI:http://dx.doi.org/10.1063/1.4905843
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:55588
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:06 Mar 2015 18:24
Last Modified:06 Mar 2015 18:24

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