Quantum chi-squared and goodness of fit testing

Journal Article · Thu Jan 15 04:00:00 EST 2015 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4905843· OSTI ID:22403085

Temme, Kristan ^[1]; Verstraete, Frank ^[2]

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1. IQIM, California Institute of Technology, Pasadena, California 91125 (United States)
2. Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria and Faculty of Science, Ghent University, B-9000 Ghent (Belgium)

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.

• OSTI ID: 22403085
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 1 Vol. 56; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DENSITY MATRIX
EIGENVALUES
HYPOTHESIS
QUANTUM MECHANICS
QUANTUM STATES
TESTING