Experimental continuous-variable cloning of partial quantum information

The fidelity of a quantum transformation is strongly linked with the prior partial information of the state to be transformed. We illustrate this interesting point by proposing and demonstrating the superior cloning of coherent states with prior partial information. More specifically, we propose two simple transformations that under the Gaussian assumption optimally clone symmetric Gaussian distributions of coherent states as well as coherent states with known phases. Furthermore, we implement for the first time near-optimal state-dependent cloning schemes relying on simple linear optics and feedforward.

Figure 1

(Color online) Schematic of the proposed $1\to 2$ cloning protocol. The signal $a_{\mathrm{in}}$ is reflected at a beam splitter with transmittance $T_1$ and detected using a beam splitter with transmittance $T_2$ and two homodyne detectors measuring the amplitude $x$ and phase $p$ quadratures. The measurement outcomes are scaled with the gains $g_x$ and $g_p$ and used to displace the transmitted signal. The displaced state is subsequently split on a symmetric beam splitter, thus producing two clones denoted by $a_{\text{clone}1}$ and $a_{\text{clone}2}$. $a_1$, $a_2$, and $a_3$ are ancilla states.

Figure 2

Experimental cloning setup. AM, amplitude modulator; PM, phase modulator; EOM, electro-optic modulator; LO, local oscillator; AUX, auxiliary state; $T_1$, variable beam splitter.

Figure 3

(Color online) Average fidelity plotted against the width $\left(\sqrt{V}\right)$ of the distribution of input states. The solid line corresponds to the theory and the red dots and black squares correspond to average fidelities for clones 1 and 2. The dashed line takes into account that the amplifier used in the scheme was nonideal and some technical noise entered the cloning process. The dotted line corresponds to a measure and prepare strategy. The gray shaded area corresponds to the region where the solution $T_1=1$ is optimal. The red line extends this solution into the region where it is not optimal.

Figure 4

(Color online) Spectral amplitude quadrature noise densities of the two clones (upper black traces) relative to the quantum noise level (lower red trace). The measurement was taken over a period of $2\mathrm{s}$. The settings of the spectrum analyzer were $14.3\mathrm{MHz}$ central frequency, $100\mathrm{kHz}$ bandwidth, and $300\mathrm{Hz}$ video bandwidth. The added noise contributions are $1.8\pm 0.1\mathrm{dB}$ and $1.85\pm 0.1\mathrm{dB}$ for clone 1 and clone 2, respectively. The optimal cloning limit ($1.75\mathrm{dB}$ above the shot noise level) is pointed out by the solid green line.