Quantum chi-squared and goodness of fit testing

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.

[1]  M. Nussbaum,et al.  Asymptotic Error Rates in Quantum Hypothesis Testing , 2007, Communications in Mathematical Physics.

[2]  Robin Blume-Kohout,et al.  Entanglement verification with finite data. , 2010, Physical review letters.

[3]  H. Weinfurter,et al.  Experimental quantum teleportation , 1997, Nature.

[4]  D. Petz,et al.  Geometries of quantum states , 1996 .

[5]  Matthias Christandl,et al.  Reliable quantum state tomography. , 2011, Physical review letters.

[6]  D. Petz.  Quantum Information Theory and Quantum Statistics , 2007 .

[7]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[8]  David Poulin,et al.  Practical characterization of quantum devices without tomography. , 2011, Physical review letters.

[9]  W. J. Langford.  Statistical Methods , 1959, Nature.

[10]  W. Wootters,et al.  Optimal state-determination by mutually unbiased measurements , 1989 .

[11]  N. Chentsov,et al.  Markov invariant geometry on manifolds of states , 1991 .

[12]  O. Barndorff-Nielsen,et al.  Fisher information in quantum statistics , 1998, quant-ph/9808009.

[13]  H. Cramér.  Mathematical methods of statistics , 1947 .

[14]  K. Audenaert,et al.  Discriminating States: the quantum Chernoff bound. , 2006, Physical review letters.

[15]  D. Petz,et al.  Contraction of Generalized Relative Entropy Under Stochastic Mappings on Matrices , 1998 .

[16]  G. Roger,et al.  Experimental Test of Bell's Inequalities Using Time- Varying Analyzers , 1982 .

[17]  C. Helstrom.  Quantum detection and estimation theory , 1969 .

[18]  M. Ruskai,et al.  Monotone Riemannian metrics and relative entropy on noncommutative probability spaces , 1998, math-ph/9808016.

[19]  G. E. Noether,et al.  ON A THEOREM OF PITMAN , 1955 .

[20]  R. R. Bahadur.  An optimal property of the likelihood ratio statistic , 1967 .

[21]  Yi-Kai Liu,et al.  Direct fidelity estimation from few Pauli measurements. , 2011, Physical review letters.

[22]  W. Wootters.  Statistical distance and Hilbert space , 1981 .

[23]  S. Braunstein,et al.  Statistical distance and the geometry of quantum states. , 1994, Physical review letters.

[24]  P. Maunz,et al.  Trapping an atom with single photons , 2000, Nature.

[25]  D. Bures.  An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite *-algebras , 1969 .

[26]  M. Nussbaum,et al.  THE CHERNOFF LOWER BOUND FOR SYMMETRIC QUANTUM HYPOTHESIS TESTING , 2006, quant-ph/0607216.

[27]  D. Petz.  Quasi-entropies for finite quantum systems , 1986 .

[28]  D. Petz.  Monotone metrics on matrix spaces , 1996 .

[29]  Michael M. Wolf,et al.  Hilbert's projective metric in quantum information theory , 2011, 1102.5170.

[30]  A. Luati.  Maximum Fisher information in mixed state quantum systems , 2004, math/0410095.

[31]  F. Schmidt-Kaler,et al.  Deterministic quantum teleportation with atoms , 2004, Nature.

[32]  A. Uhlmann.  The "transition probability" in the state space of a ∗-algebra , 1976 .

[33]  Masahito Hayashi,et al.  Quantum Information: An Introduction , 2010 .

[34]  Wataru Kumagai,et al.  Quantum Hypothesis Testing for Gaussian States: Quantum Analogues of χ2, t-, and F-Tests , 2011, 1110.6255.

[35]  F. Verstraete,et al.  The χ2-divergence and mixing times of quantum Markov processes , 2010, 1005.2358.

[36]  G. W. Snedecor.  Statistical Methods , 1964 .

[37]  Kimble,et al.  Unconditional quantum teleportation , 1998, Science.

[38]  F. Hiai,et al.  The proper formula for relative entropy and its asymptotics in quantum probability , 1991 .

[39]  H. Yuen,et al.  Review of 'Quantum Detection and Estimation Theory' (Helstrom, C. W.; 1976) , 1977, IEEE Transactions on Information Theory.

[40]  King,et al.  Demonstration of a fundamental quantum logic gate. , 1995, Physical review letters.