Complete population inversion of maximally entangled states in 22N -level systems via Pythagorean-triple coupling

Maximally entangled states play a central role in quantum information processing. Despite much progress throughout the years, robust protocols for manipulations of such states in many-level systems are still scarce. Here we present a control scheme that allows efficient manipulation of complete population inversion between two maximally entangled states. Exploiting the self-duality of SU(2), we present in this work a family of 22N-level systems with couplings related to Pythagorean triples that make a complete population inversion from one state to another (orthogonal) state, using very few couplings and generators. We relate our method to the recently developed retrograde-canon scheme and derive a more general complete transfer recipe. We also discuss the cases of (2n)2-level systems, (2n+1)2-level systems, and other unitary groups, and give a geometrical description of the inversion via the Majorana sphere.