Optimal probabilistic cloning and purification of quantum states

We investigate the probabilistic cloning and purification of quantum states. The performance of these probabilistic operations is quantified by the average fidelity between the ideal and actual output states. We provide a simple formula for the maximal achievable average fidelity and we explicitly show how to construct a probabilistic operation that achieves this fidelity. We illustrate our method on several examples such as the phase covariant cloning of qubits, cloning of coherent states, and purification of qubits transmitted via depolarizing channel and amplitude damping channel. Our examples reveal that the probabilistic cloner may yield higher fidelity than the best deterministic cloner even when the states that should be cloned are linearly dependent and are drawn from a continuous set.

Figure 1

The maximal possible improvements of fidelity $\Delta F\left(\eta \right)$ (solid line) and $\Delta F_{\perp }\left(\eta \right)$ (dashed line) that can be achieved by purification when the two-qubit state $\mid \psi \psi ⟩$ or $\mid \psi {\psi }_{\perp }⟩$, respectively, is sent through the depolarizing channel with parameter $\eta$.

Figure 2

(a) Maximal fidelity of the purified qubit when Alice sends $N$ copies of the qubit via amplitude damping channel parametrized by $\eta$. For comparison, the curve labeled $F_0$ displays the fidelity of the single qubit after passing through the channel, $F_0=(1+\eta )∕2.$ (b) The corresponding maximal probability of successful purification.