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

Abstract

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 $\mathrm{SU}(2)$, we present in this work a family of ${2}^{2N}$-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.