  In this paper, we study $k$-positivity and Schmidt number under standard
orthogonal group symmetries. The Schmidt number is a natural quantification of
entanglement in quantum information theory. First of all, we exhibit a complete
characterization of all orthogonally covariant $k$-positive maps. This
generalizes earlier results in [Tom85]. Furthermore, we optimize duality
relations between $k$-positivity and Schmidt numbers under compact group
symmetries. This new framework enables us to efficiently compute the Schmidt
numbers of all orthogonally invariant quantum states.
