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Keywords:

  • Likelihood ratio test;
  • Maximum likelihood estimation;
  • Multinomial distribution;
  • Poisson distribution;
  • Quantum mechanics;
  • Qubit;
  • Score test

Summary.  The simplest states of finite quantum systems are the pure states. The paper is motivated by the need to test statistically whether or not a given physical state is pure. Because the pure states lie in the boundary of the set of all states, the usual regularity conditions that justify the standard large sample approximations to the null distributions of the deviance and the score statistic are not satisfied. For a large class of quantum experiments that produce Poisson count data, the paper uses an enlargement of the parameter space of all states to develop likelihood ratio and score tests of purity. The asymptotic null distributions of the corresponding statistics are χ2. The tests are illustrated by the analysis of some quantum experiments involving unitarily correctable codes.