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

TL;DR

This paper investigates how nonnegativity constraints affect estimation errors in one-qubit state tomography with finite data, revealing significant deviations from asymptotic predictions and providing a practical method to determine data sufficiency.

Contribution

It introduces an explicit function modeling finite-data estimation errors considering boundary effects, enhancing understanding and reliability of quantum state tomography.

Findings

Large gap between finite-data errors and asymptotic predictions.

Derived an explicit error approximation function with high accuracy.

Provided a formula for data amount needed for reliable estimation.

Abstract

We analyze the behavior of estimation errors evaluated by two loss functions, 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 the boundary in the state space imposed by the requirement that density matrices be nonnegative (positive semidefinite). We derive an explicit form of a function reproducing the behavior of the estimation errors with high accuracy by introducing two approximations: a Gaussian approximation of the multinomial distributions of outcomes, and linearizing the boundary. This function gives us an intuition for 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.