Information geometry of randomized quantum state tomography

Abstract

Suppose that a d-dimensional Hilbert space H ≃ Cd 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 measurements M(a) = {|1(a)〉 〈1(a)|, . . ., |d(a)〉 〈d(a)|} for a = 1, . . ., d + 1, 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.