Information Geometry of Randomized Quantum State Tomography

Suppose that a d-dimensional Hilbert space [Formula: see text] admits a full set of mutually unbiased bases [Formula: see text] , where [Formula: see text]. A randomized quantum state tomography is a scheme for estimating an unknown quantum state on [Formula: see text] through iterative applications of measurements [Formula: see text] for [Formula: see text] , where the numbers of applications of these measurements are random variables. We show that the space of the resulting probability distributions enjoys a mutually orthogonal dualistic foliation structure, which provides us with a simple geometrical insight into the maximum likelihood method for the quantum state tomography.