Exponential advantage in continuous-variable quantum state learning

We consider the task of learning quantum states in bosonic continuous-variable (CV) systems. We present an experimentally feasible protocol that uses entangled measurements and reflected states to efficiently learn, up to sign, the characteristic function of CV quantum states, with sample complexity independent of the number of modes $n$. We prove that any adaptive scheme without entangled measurements requires exponentially many samples in $n$ for this learning task, thereby demonstrating an exponential advantage from entangled measurements. Remarkably, we also prove that any entanglement-assisted scheme that does not have access to reflected states requires exponentially many samples in $n$. Together, these results establish a rigorous exponential advantage that jointly relies on entangled measurements and reflected states. Finally, we relate the sample-complexity bounds to the input state's classicality, revealing a classicality-complexity tradeoff: entanglement-free schemes suffice for highly classical states, whereas in the genuinely nonclassical regime the sample cost can be reduced only by providing both entanglement and reflected states.