Error probability analysis in quantum tomography: A tool for evaluating experiments

We expand the scope of the statistical notion of error probability, that is, how often large deviations are observed in an experiment, to make it directly applicable to quantum tomography. We verify that the error probability can decrease at most exponentially in the number of trials, we derive the explicit rate that bounds this decrease, and we show that a maximum likelihood estimator achieves this bound. We also show that the statistical notion of identifiability coincides with the tomographic notion of informational completeness. Our result implies that two quantum tomographic apparatuses that have the same risk function (e.g., variance) can have different error probability, and we give an example in one-qubit-state tomography. Thus by combining these two approaches we can evaluate, in a reconstruction-independent way, the performance of such experiments more discerningly.