We initiate the study of sample-optimal quantum state tomography with minimal disturbance to the samples. Can we efficiently learn a precise description of a quantum state through sequential measurements of samples while at the same time making sure that the post-measurement state of the samples is only minimally perturbed? Defining regret as the cumulative disturbance of all samples, the challenge is to find a balance between the most informative sequence of measurements on the one hand and measurements incurring minimal regret on the other. Here we answer this question for qubit states by exhibiting a protocol that for pure states achieves maximal precision while incurring a regret that grows only polylogarithmically with the number of samples, a scaling that we show to be optimal.
@article{arxiv.2406.18370,
title = {Learning pure quantum states (almost) without regret},
author = {Josep Lumbreras and Mikhail Terekhov and Marco Tomamichel},
journal= {arXiv preprint arXiv:2406.18370},
year = {2025}
}