  For every possible spectrum of $2^N$-dimensional density operators, we
construct an $N$-qubit X-state of same spectrum and maximal genuine
multipartite (GM-) concurrence, hence characterizing a global unitary
transformation that --- constrained to output X-states --- maximizes the
GM-concurrence of an arbitrary input mixed state of $N$ qubits. We also apply
semidefinite programming methods to obtain $N$-qubit X-states with maximal
GM-concurrence for a given purity and to provide an alternative proof of
optimality of a recently proposed set of density matrices for the role, the
so-called X-MEMS. Furthermore, we introduce a numerical strategy to tailor a
quantum operation that converts between any two given density matrices using a
relatively small number of Kraus operators. We apply our strategy to design
short operator-sum representations for the transformation between any given
$N$-qubit mixed state and a corresponding X-MEMS of same purity.

Quantum Physics