Online Learning of Pure States is as Hard as Mixed States

TL;DR

This paper demonstrates that in the online learning framework, learning pure quantum states is as computationally hard as learning mixed states, challenging previous assumptions about their relative complexity.

Contribution

It proves that the complexity of learning pure states matches that of mixed states in online quantum learning, using the concept of sequential fat-shattering dimension.

Findings

Pure and mixed states share similar regret scaling in online learning.

The results extend to the $\\epsilon$-realizable setting and partial density matrix learning.

The study generalizes previous quantum state tomography results to new settings.

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.