Optimal cloning of qubits given by an arbitrary axisymmetric distribution on the Bloch sphere

We find an optimal quantum cloning machine, which clones qubits of arbitrary symmetrical distribution around the Bloch vector with the highest fidelity. The process is referred to as phase-independent cloning in contrast to the standard phase-covariant cloning for which an input qubit state is a priori better known. We assume that the information about the input state is encoded in an arbitrary axisymmetric distribution (phase function) on the Bloch sphere of the cloned qubits. We find analytical expressions describing the optimal cloning transformation and fidelity of the clones. As an illustration, we analyze cloning of qubit state described by the von Mises–Fisher and Brosseau distributions. Moreover, we show that the optimal phase-independent cloning machine can be implemented by modifying the mirror phase-covariant cloning machine for which quantum circuits are known.

Figure 1

Qubit states described on the Bloch sphere by the von Mises–Fisher distribution (where the symmetry axis is rotated from the north pole by $\vartheta =\pi /3$ and $\phi =\pi /35$) divided by its largest value for (a) $\kappa =0$, (b) $\kappa =0.3305$, (c) $\kappa =1$, and (d) $\kappa =10$. The darker the region, the higher the values of the probability distribution function. The mean value $\langle {\sigma }_z\rangle =1$. The optimal cloning machines are the UC [2] for case (a) and the PCC [9] for the other cases. Note that the transition area between (a) and (b) corresponds neither to the UC nor the PCC.

Figure 2

Average single-copy fidelity $F$ vs. concentration $\kappa$. The dashed line for $F=5/6$ corresponds to the UC limit, which is reached for $\kappa =0$. The shaded area corresponds to the range $-0.3305<\kappa <0.3305$, where the PCC is no longer optimal.

Figure 3

Average single-clone fidelity $F$ vs. parameter $\mu$ for ${\mu }^2=P^2$ (solid) and $\mu \ne P=1$ (dashed curve). The dotted line for $F=5/6$ shows the UC limit, which is reached for $\mu =P=0$, whereas the dashed curve corresponds to the fidelity of the PCC.

Figure 4

Same as Fig. 1, but for the Brosseau distribution assuming (a) $\mu =0$, $P=0.99$, (b) $\mu =0$, $P=0.65$, (c) $\mu =0.8$, $P=0.99$, and (d) $\mu =0.8$, $P=0.8$. The optimal cloning machines are the MPCC [12] with $\Lambda =\cos {\alpha }_{+}$ for case (a) and the PCC [9] for the other cases.

Figure 5

A quantum circuit which implements the optimal phase-independent cloning transformation of $|\psi (\theta ,\varphi )\rangle =\cos (\theta /2)|0\rangle +\exp (i\varphi )\sin (\theta /2)|1\rangle$ described on the Bloch sphere by arbitrary axisymmetric distribution function $g(\theta )$. From left to right: controlled rotation $R_y$ about the $y$ axis by angle $\Phi =2({\alpha }_{-}-{\alpha }_{+})$, rotation $R_y$ by angle $\omega =2{\alpha }_{+}$, controlled Hadamard gate, and three cnot gates.

Figure 6

A quantum circuit, shown in Fig. 5, implementing the optimal phase-independent cloning transformation, is equivalent to the MPCC [12] with $\Lambda =\cos {\alpha }_{+}$ together with the controlled rotation $R_y$ by angle $\Phi =2({\alpha }_{-}-{\alpha }_{+})$.