Information Geometry of Randomized Quantum State Tomography

Abstract

Suppose that a d-dimensional Hilbert space H C d admits a full set of mutually unbiased bases |1 (a) , . . . , |d (a) , where a = 1, . . . , d + 1. A randomized quantum state tomography is a scheme for estimating an unknown quantum state on H through iterative applications of measurementswhere 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.