Precision-guaranteed quantum tomography

TL;DR

This paper introduces a rigorous estimator for quantum state tomography that guarantees the precision of state preparation, providing explicit probability bounds applicable to various measurement schemes and loss functions.

Contribution

It presents a new estimator with proven high-probability closeness to the true state, enabling rigorous evaluation of quantum state preparation accuracy.

Findings

Derived explicit probability bounds for estimator accuracy

Applicable to any informationally complete measurement set

Works for multiple data sets and various loss functions

Abstract

Quantum state tomography is the standard tool in current experiments for verifying that a state prepared in the lab is close to an ideal target state, but up to now there were no rigorous methods for evaluating the precision of the state preparation in tomographic experiments. We propose a new estimator for quantum state tomography, and prove that the (always physical) estimates will be close to the true prepared state with high probability. We derive an explicit formula for evaluating how high the probability is for an arbitrary finite-dimensional system and explicitly give the one- and two-qubit cases as examples. This formula applies for any informationally complete sets of measurements, arbitrary finite number of data sets, and general loss functions including the infidelity, the Hilbert-Schmidt, and the trace distances. Using the formula, we can evaluate not only the difference between the estimated and prepared states, but also the difference between the prepared and target states. This is the first result directly applicable to the problem of evaluating the precision of estimation and preparation in quantum tomographic experiments.