k-Positivity and Schmidt number under orthogonal group symmetries

In this paper, we present a new application of group theory to develop a systematical approach to efficiently compute the Schmidt numbers. The Schmidt number is a natural quantification of entanglement in quantum information theory, but computing its exact value is generally a challenging task even for very concrete examples. We exhibit a complete characterization of all orthogonally covariant k -positive maps. This result generalizes earlier results by Tomiyama (Linear Algebra Appl 69:169–177, 1985). Furthermore, we optimize duality relations between k -positivity and Schmidt numbers under group symmetries. This new approach enables us to transfer the results of k -positivity to the computation of the Schmidt numbers of all orthogonally invariant quantum states.