Online Learning of Pure States is as Hard as Mixed States

Maxime Meyer, Soumik Adhikary, Naixu Guo, Patrick Rebentrost

Keywords: quantum state learning, tomography, pure and mixed states, online learning, sequential fat-shattering dimension, smoothed analysis

TL;DR: We study online learning of quantum states and prove that learning a pure state is as hard as learning a mixed state, id est that the rank of the matrix has no impact on the complexity of the learning problem.

Abstract: Quantum state tomography, the task of learning an unknown quantum state, is a fundamental problem in quantum information. In standard settings, the complexity of this problem depends significantly on the type of quantum state that one is trying to learn, with pure states being substantially easier to learn than general mixed states. A natural question is whether this separation holds for any quantum state learning setting. In this work, we consider the online learning framework and prove the surprising result that learning pure states in this setting is as hard as learning mixed states. More specifically, we show that both classes share almost the same sequential fat-shattering dimension, leading to identical regret scaling. We also generalize previous results on full quantum state tomography in the online setting to (i) the $\epsilon$-realizable setting and (ii) learning the density matrix only partially, using smoothed analysis.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 25944