We characterize learnability for quantum measurement classes by establishing matching necessary and sufficient conditions for their PAC learnability, along with corresponding sample complexity bounds, in the setting where the learner is given access only to prepared quantum states. We first probe the results from previous works on this setting. We show that the empirical risk defined in previous works and matching the definition in the classical theory fails to satisfy the uniform convergence property enjoyed in the classical setting for some learnable classes. Moreover, we show that VC dimension generalization upper bounds in previous work are frequently infinite, even for finite-dimensional POVM classes. To surmount the failure of the standard ERM to satisfy uniform convergence, we define a new learning rule -- denoised ERM. We show this to be a universal learning rule for POVM and probabilistically observed concept classes, and the condition for it to satisfy uniform convergence is finite fat shattering dimension of the class. We give quantitative sample complexity upper and lower bounds for learnability in terms of finite fat-shattering dimension and a notion of approximate finite partitionability into approximately jointly measurable subsets, which allow for sample reuse. We then show that finite fat shattering dimension implies finite coverability by approximately jointly measurable subsets, leading to our matching conditions. We also show that every measurement class defined on a finite-dimensional Hilbert space is PAC learnable. We illustrate our results on several example POVM classes.

翻译：我们通过建立量子测量类别的PAC可学习性的匹配必要与充分条件，并给出相应的样本复杂度界，刻画了学习者在仅能访问制备量子态场景下的可学习性。首先，我们深入探讨了先前工作在该设置下的结果。研究表明，先前工作中定义的经验风险（与经典理论中的定义相匹配）对于某些可学习类别而言，未能满足经典设置中普遍适用的一致性收敛性质。此外，我们发现先前工作中基于VC维的泛化上界经常是无穷大的，即便对于有限维的POVM类别也是如此。为解决标准经验风险最小化无法满足一致收敛性的问题，我们定义了一种新的学习规则——去噪经验风险最小化。我们证明了该规则是POVM类及概率观测概念类别的通用学习规则，且其满足一致收敛性的条件是该类别的有限fat shattering维。我们给出了基于有限fat shattering维和近似联合可测子集近似有限可划分性的定量样本复杂度上界与下界（该划分允许样本重用）。随后论证了有限fat shattering维意味着可通过近似联合可测子集实现有限覆盖，从而导出匹配条件。我们还证明了定义在有限维希尔伯特空间上的任意测量类别都是PAC可学习的。最后通过若干POVM类别实例验证了所得结论。