LOCC convertibility of entangled states in infinite-dimensional systems

We advance on the conversion of bipartite quantum states via local operations and classical communication for infinite-dimensional systems. We introduce δ-convertibility based on the observation that any pure state can be approximated by a state with finite-support Schmidt coefficients. We show that δ-convertibility of bipartite states is fully characterized by a majorization relation between the sequences of squared Schmidt coefficients, providing a novel extension of Nielsen's theorem for infinite-dimensional systems. Hence, our definition is equivalent to the one of ε-convertibility [Quantum Inf. Comput. \textbf{8}, 0030 (2008)], but deals with states having finitely supported sequences of Schmidt coefficients. Additionally, we discuss the notions of optimal common resource and optimal common product in this scenario. The optimal common product always exists, whereas the optimal common resource depends on the existence of a common resource. This highlights a distinction between the resource-theoretic aspects of finite versus infinite-dimensional systems. Our results rely on the order-theoretic properties of majorization for infinite sequences, applicable beyond the LOCC convertibility problem.