Complete Population Transfer of Maximally Entangled States in $2^{2N}$-level Systems via Pythagorean Triples 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 allow efficient manipulation of complete population transfer between two maximally entangled states. Exploiting the self-duality of $\mathrm{SU}\left(2\right)$, we present in this work a family of ${\mathrm{2}}^{\mathrm{2}N}$-level systems with couplings related to Pythagorean triples that make a complete population transfer 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 $\left(2n\right)^2$-level systems, $\left(2n+1\right)^2$-level systems and other unitary groups.