Learning pure quantum states (almost) without regret
This addresses the challenge of sample-optimal quantum state tomography for qubit states, representing an incremental advance in balancing measurement informativeness and minimal regret.
The paper tackles the problem of learning pure quantum states with minimal disturbance to the samples, achieving maximal precision while incurring regret that grows only polylogarithmically with the number of samples, which is shown to be optimal.
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.