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

Abstract

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.

22 pages, 2 figures. v2 (published version): connection with "biseparability from spectrum" dropped from the concluding remarks since, in the last paragraph os Sec. 3, an argument was included to demonstrate that N-qubit X-MEMS are generally not N-qubit MEMS

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