Economical phase-covariant cloning of qudits

We derive the optimal $N\to M$ phase-covariant quantum cloning for equatorial states in dimension $d$ with $M=kd+N$, $k$ integer. The cloning maps are optimal for both global and single-qudit fidelity. The map is achieved by an “economical” cloning machine, which works without ancilla.

Figure 1

Single-qubit fidelity for $1\to M$ cloning: phase-covariant (continuous line); universal (dotted line). The numerical results are equal to those in Refs. 5, 8.

Figure 2

Single-qutrit fidelity for $1\to M$ cloning: multiphase (continuous line); universal (dotted line).

Figure 3

Single-qudit, $d=5$, fidelity for $1\to M$ cloning: multiphase (continuous line); universal (dotted line).

Figure 4

Comparison between single-qudit fidelities for multiphase covariant $1\to M$ cloning: $d=2$ (continuous line); $d=3$ (dotted line); $d=5$ (dash-dotted line). The three curves for large $M$ approach the fidelity of optimal phase estimation over one copy, namely $F=3∕4$ for qubits, $F=5∕9$ for qutrits, and $F=9∕25$ for $d=5$.

Figure 5

Saturation of single-qubit fidelity for phase-covariant (continuous line) and universal (dotted line) $N\to M$ cloning machines versus $N$, for $M=21$.

Figure 6

Saturation of single-qutrit fidelity for multiphase (continuous line) and universal (dotted line) $N\to M$ cloning machines versus $N$, for $M=28$.

Figure 7

Saturation of single-qudit, $d=5$, fidelity for multiphase (continuous line) and universal (dotted line) $N\to M$ cloning machines versus $N$, for $M=51$.