在线学习纯态与混态一样难，突破了传统认知。
Online Learning of Pure States is as Hard as Mixed States
- 在线学习框架下，纯态与混态的复杂度一致
- 两者序列胖破碎维数几乎相同，后悔值增长速率相同
- 适用于量子态学习、在线算法研究者
量子态层析，即学习未知量子态的任务，是量子信息中的基础问题。在标准设定下，纯态的学习复杂度远低于一般混态。一个自然问题是：这种差异是否在所有量子态学习框架中都存在？本文研究在线学习框架，证明了一个令人惊讶的结果：在此设定下，学习纯态与学习混态一样困难。具体而言，我们表明两类状态具有几乎相同的序列胖破碎维数，导致后悔值增长速率一致。此外，我们将以往关于全量子态层析在线学习的结果推广至（i）ε可实现情形，以及（ii）使用平滑分析部分学习密度矩阵的情形。
原文摘要 · Abstract (English)
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.
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