LOCC convertibility of entangled states in infinite-dimensional systems

Abstract

We advance on the conversion of bipartite quantum states via local operations and classicalcommunication (LOCC) for infinite-dimensional systems. We introduce δ-LOCC convertibilitybased on the observation that any pure state can be approximated by a state with finite-supportSchmidt coefficients. We show that δ-LOCC convertibility of bipartite states is fully characterizedby a majorization relation between the sequences of squared Schmidt coefficients, providing a novelextension of Nielsen’s theorem for infinite-dimensional systems. Hence, our definition is equivalentto the one of ϵ-LOCC convertibility (Owari et al 2008 Quantum Inf. Comput. 8 0030), but dealswith states having finitely supported sequences of Schmidt coefficients. Additionally, we discuss thenotions of optimal common resource and optimal common product in this scenario. The optimalcommon product always exists, whereas the optimal common resource depends on the existence ofa common resource. This highlights a distinction between the resource-theoretic aspects of finiteversus infinite-dimensional systems. Our results rely on the order-theoretic properties ofmajorization for infinite sequences, applicable beyond the LOCC convertibility problem.