Learning pure quantum states (almost) without regret

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{lumbreras2025_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 }
}