Online Learning of Pure States is as Hard as Mixed States

Maxime Meyer (National University of Singapore) · Soumik Adhikary (Centre for Quantum Technologies, National University of Singapore) · Naixu Guo (National University of Singapore) · Patrick Rebentrost (National University of Singapore)

$\epsilon$-realizable settingcomplexity theorydensity matrixgeneralization learning theorymixed statesonline learning frameworkpartial learningpure statesquantum informationquantum learningquantum state tomographyregret scalingsequential fat-shattering dimensionsmoothed analysisstate learning

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.