Counterfactual quantum cloning without transmitting any physical particles

We propose a counterfactual $1\to 2$ economical phase-covariant cloning scheme. Compared with the existing protocols using flying qubits, the main difference of the presented scheme is that the cloning can be achieved without transmitting the photon between the two parties. In addition, this counterfactual scheme does not need to construct controlled quantum gates to perform joint logical operations between the cloned qubit and the blank copy. We also numerically evaluate the performance of the present scheme in the practical experiment, which shows this cloning scheme can be implemented with a high success of probability and the fidelity is close to the optimal value in the ideal asymptotic limit.

Figure 1

The quantum control device for the photon path. MR: normal mirror; BS: 50:50 beam splitter; D: conventional photon detector; ${\mathrm{CM}}_1$ and ${\mathrm{CM}}_2$ compose a single-side cavity.

Figure 2

Schematic of the counterfactual $1\to 2$ economical phase-covariant cloning. ${\mathrm{PBS}}_k$ ($k=1,2,3$): polarizing beam splitter; ${\mathrm{SM}}_{1\left(2\right)}$: switchable mirror; ${\mathrm{SPR}}_{1\left(2\right)}$: switchable polarization rotator, where the arrow means ${\mathrm{SPR}}_{1\left(2\right)}$ can rotate only the photon coming from the ${\mathrm{SM}}_{1\left(2\right)}$ side; ${\mathrm{D}}_{1\left(2\right)}$: conventional photon detector; ${\mathrm{HWP}}_{1\left(3\right)}$: half-wave plate oriented at ${45}^{\circ }$; ${\mathrm{HWP}}_2$: half-wave plate oriented at $22.5^{\circ }$; PS: phase shifter; BS: 50:50 beam splitter; OD: optical delay line.

Figure 3

The parameters $x_M-y_M+z_M$, $x_M+y_M-z_M$, $x_M-y_M-z_M$, and $x_M+y_M+z_M$ in Eq. (13) versus the different values of $N$ and $M$. (a) $x_M-y_M+z_M$ is close to 2, (b) $x_M+y_M-z_M$ approaches to 0, (c) $x_M-y_M-z_M$ is close to 0, (d) $x_M+y_M+z_M$ tends to 2 for large $N$ and $M$.

Figure 4

The average success probability (a) and the average fidelity (b) of the counterfactual cloning scheme versus different values of outer and inner cycles $M$ and $N$.

Figure 5

The average successful probability (solid lines) and the average fidelity (dotted lines) versus the error coefficient $s$ of the SPR for the different values of $M$ and $N$.

Figure 6

The average successful probability (solid lines) and the average fidelity (dotted lines) versus the photon loss probability $\gamma$ in the transmission channel for the different values of $M$ and $N$.