Optimal manipulations with qubits: Universal quantum entanglers

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We analyze various scenarios for entangling two initially unentangled qubits. In particular, we propose an optimal universal entangler which entangles a qubit in unknown state |Ψ> with a qubit in a reference (known) state |0>. That is, our entangler generates the output state which is as close as possible to the pure (symmetrized) state (|Ψ>|0> +|0>|Ψ>). The most attractive feature of this entangling machine, is that the fidelity of its performance (i.e. the distance between the output and the ideally entangled -- symmetrized state) does not depend on the input and takes the constant value F= (9+3√2)/14≃ 0.946. We also analyze how to optimally generate from a single qubit initially prepared in an unknown state |Ψ a two qubit entangled system which is as close as possible to a Bell state (|Ψ|Ψ⊥+|Ψ⊥|Ψ), where łΨ|Ψ⊥ =0.