On the Chernoff bound for efficiency of quantum hypothesis testing

The quantum statistic is a rapidly growing area of modern statistics (see [1] and [2]). Why? Today, quantum states can be manufactured. For example, in one method ions are placed in a trap created by electrostatic potential and radio-frequency oscillations. The ions then are cooled by laser emission, and arranged on a line in the trap. After that, each individual ion can be accessed by laser pulses and their joint quantum state can be altered according to the researcher’s wishes. This ability to built and manipulate quantum systems is changing our thinking about computation and information transmission. Suddenly, certain classic problems — the factorization of large integers, the search in an unstructured database, secure communication — are not as difficult as they always were. This conceptual change also affects statistics. For example, how can a quantum state manufacturer check if states have been generated faithfully? We can anticipate the statistician’s answer: Select a sample of the states and perform a statistical test. But now, besides designing the test, the statistician must play an additional role, the role of advisor on how to perform measurements of a sample of quantum states. Since in quantum mechanics both the measurement and the state determine the probability distribution of outcomes, the choice of measurement affects the properties of the statistical test. The paper contributes by estimating the Chernoff-Hoeffding rate for the efficiency of testing hypotheses about quantum states