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Maximally entangled states play a central role in quantum informationprocessing. Despite much progress throughout the years, robust protocols formanipulations of such states in many-level systems are still scarce. Here wepresent a control scheme that allow efficient manipulation of completepopulation transfer between two maximally entangled states. Exploiting theself-duality of $\mathrm{SU}\left(2\right)$, we present in this work a familyof ${\mathrm{2}}^{\mathrm{2}N}$-level systems with couplings related toPythagorean triples that make a complete population transfer from one state toanother (orthogonal) state, using very few couplings and generators. We relateour method to the recently-developed retrograde-canon scheme and derive a moregeneral complete transfer recipe. We also discuss the cases of$\left(2n\right)^2$-level systems, $\left(2n+1\right)^2$-level systems andother unitary groups.

Complete Population Transfer of Maximally Entangled States in $2^{2N}$-level Systems via Pythagorean Triples Coupling