Effect of non-negativity on estimation errors in one-qubit state tomography with finite data

Abstract

We analyze the behavior of estimation errors evaluated by two loss functions, namely the Hilbert–Schmidt distance and infidelity, in one-qubit state tomography with finite data. We show numerically that there can be a large gap between the estimation errors and those predicted by an asymptotic analysis. The origin of this discrepancy is the existence of a boundary in the state space imposed by the requirement that density matrices be non-negative (positive semidefinite). We derive the explicit form of a function reproducing the behavior of estimation errors with high accuracy by introducing two approximations: a Gaussian approximation of the multinomial distributions of outcomes and a linearization of the boundary. This function gives us an intuition of the behavior of the expected losses for finite data sets. We show that this function can be used to determine the amount of data necessary for the estimation to be treated reliably with the asymptotic theory. We give an explicit expression for this amount, which exhibits strong sensitivity to the true quantum state as well as the choice of measurement.