Advantage of Entangled Learning Rules in Quantum Measurement Class Learning

Learning with data in the form of quantum states is of current interest and has led to a variety of problems that boil down to interaction with the available data via quantum measurement and classical post-processing of observed classical outcomes. In quantum measurement PAC learning, one is given a sequence of unknown, prepared quantum states and classical labels, along with a hypothesis class of candidate measurements. The task is to select a measurement from the hypothesis class that minimizes a fixed notion of error in prediction of the classical labels via measurement of a new state by the selected hypothesis. In this work, we consider the advantage of interacting with the given data in the measurement learning framework using learning rules given by measurements that cannot be implemented using local operations and classical communication (LOCC), as opposed to single-copy learning rules. We provide a construction showing that there exist learning scenarios wherein single-copy learning rules are asymptotically suboptimal compared to optimal ones. We then show that learning rules based on entangled measurements enjoy at most a polynomial sample complexity advantage over single-copy learning rules in the PAC learning setting (under a natural joint measurability covering assumption).