Maximally genuine multipartite entangled mixed X-states of N-qubits

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For every possible spectrum of -dimensional density operators, we construct an N-qubit X-state of the 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 purpose, 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 the same purity.