The density matrix in quantum mechanics parameterizes the statistical
properties of the system under observation, just like a classical probability
distribution does for classical systems. The expectation value of observables
cannot be measured directly, it can only be approximated by applying classical
statistical methods to the frequencies by which certain measurement outcomes
(clicks) are obtained. In this paper, we make a detailed study of the
statistical fluctuations obtained during an experiment in which a hypothesis is
tested, i.e. the hypothesis that a certain setup produces a given quantum
state. Although the classical and quantum problem are very much related to each
other, the quantum problem is much richer due to the additional optimization
over the measurement basis. Just as in the case of classical hypothesis
testing, the confidence in quantum hypothesis testing scales exponentially in
the number of copies. In this paper, we will argue 1) that the physically
relevant data of quantum experiments is only contained in the frequencies of
the measurement outcomes, and that the statistical fluctuations of the
experiment are essential, so that the correct formulation of the conclusions of
a quantum experiment should be given in terms of hypothesis tests, 2) that the
(classical) χ2 test for distinguishing two quantum states gives rise to
the quantum χ2 divergence when optimized over the measurement basis, 3)
present a max-min characterization for the optimal measurement basis for
quantum goodness of fit testing, find the quantum measurement which leads both
to the maximal Pitman and Bahadur efficiency, and determine the associated
divergence rates.Comment: 22 Pages, with a new section on parameter estimatio
