Optimal manipulations with qubits: Universal quantum entanglers

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 $|\Psi>$ 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 $(|\Psi>|0> +|0>|\Psi>)$. 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 $F= (9+3\sqrt{2})/14\simeq 0.946$. We also analyze how to optimally generate from a single qubit initially prepared in an unknown state $|\Psi\r$ a two qubit entangled system which is as close as possible to a Bell state $(|\Psi\r|\Psi^\perp\r+|\Psi^\perp\r|\Psi\r)$, where $\l\Psi|\Psi^\perp\r =0$.