Learning pure quantum states (almost) without regret

TL;DR

This paper introduces a quantum state tomography protocol that efficiently learns pure qubit states with minimal sample disturbance, achieving near-optimal precision and low cumulative measurement regret.

Contribution

It presents a novel measurement protocol for pure quantum states that balances high accuracy with minimal sample disturbance, optimizing the trade-off between information gain and regret.

Findings

Achieves maximal precision in pure state tomography.

Regret grows polylogarithmically with the number of samples.

Proves the optimality of the regret scaling.

Abstract

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.