Computing entanglement costs of non-local operations on the basis of algebraic geometry

In the study of distributed quantum information processing, it is crucial to minimize the entanglement consumption by optimizing local operations and classical communications (LOCC). We develop a framework based on algebraic geometry to systematically simplify the optimization over separable (SEP) channels, providing a good approximation of LOCC. As a first application, we generalize previous results on the one-shot entanglement cost of non-local operations in a unified way. Via the generalization, we also resolve an open problem posed by Yu et al. regarding the entanglement cost of local state discrimination. As a second application, we strengthen the Doherty--Parrilo--Spedalieri hierarchy and numerically determine the trade-off between the entanglement cost and the success probability of implementing various non-local operations under separable channels---such as entanglement distillation, non-local unitary channels, measurements, and state verification.