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We consider translationally invariant states of an infinite one dimensionalchain of qubits or spin-1/2 particles. We maximize the entanglement shared bynearest neighbours via a variational approach based on finitely correlatedstates. We find an upper bound of nearest neighbour concurrence equal toC=0.434095 which is 0.09% away from the bound C_W=0.434467 obtained by acompletely different procedure. The obtained state maximizing nearest neighbourentanglement seems to approximate the maximally entangled mixed states (MEMS).Further we investigate in detail several other properties of the so obtainedoptimal state.Comment: 12 pages, 4 figures, 2nd version minor change

Maximizing nearest-neighbor entanglement in finitely correlated qubit chains