Online Learning of Pure States is as Hard as Mixed States
This result challenges the assumption that pure states are easier to learn than mixed states in quantum information, potentially impacting quantum computing and machine learning applications.
The paper tackles the problem of quantum state tomography in an online learning framework, showing that learning pure states is as hard as learning mixed states, with both classes having nearly identical sequential fat-shattering dimensions and regret scaling.
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 $ε$-realizable setting and (ii) learning the density matrix only partially, using smoothed analysis.