Optimal multi-setting witness operators for two qudits

Quantum entanglement in bipartite systems is approached with the help of pseudo Bell operators, a relaxed form of Bell operators. Their connection to optimal witness operators is established. A method to construct pseudo Bell operators for pairs of qudits, based on multiple two-dimensional local measuring settings, is presented. Each of them provide two optimal (multi-setting) witness operators. They are rigorously proved to fulfill the conditions in the definition. We explain how to approximate the set of separable states and the set of entangled states. A class of Werner-like entangled states is identified. The spectral resolution of all multi-setting pseudo Bell operators is carried out in full.