Cloning of Gaussian states by linear optics

We analyze in details a scheme for cloning of Gaussian states based on linear optical components and homodyne detection recently demonstrated by Andersen et al. [Phys. Rev. Lett. 94, 240503 (2005)]. The input-output fidelity is evaluated for a generic (pure or mixed) Gaussian state taking into account the effect of nonunit quantum efficiency and unbalanced mode mixing. In addition, since in most quantum information protocols the covariance matrix of the set of input states is not perfectly known, we evaluate the average cloning fidelity for classes of Gaussian states with the degree of squeezing and the number of thermal photons being only partially known.

Figure 1

Cloning of Gaussian states by linear optics: the input state ${\varrho }_{\mathrm{in}}$ is mixed with the vacuum ${\varrho }_0$ at a beam splitter (BS) of transmissivity ${\tau }_1$. The reflected beam is measured by double-homodyne detection and the outcome of the measurement $x+iy$ is forwarded to a modulator, which imposes a displacement $g(x+iy)$ on the transmitted beam, $g$ being a suitable amplification factor. Finally, the displaced state is impinged onto a second beam splitter of transmissivity ${\tau }_2$. The two outputs, ${\varrho }_1$ and ${\varrho }_2$, from the beam splitter represent the two clones.

Figure 2

Linear cloning fidelity $F_{\eta }$ for a coherent state as a function of the BS transmittivity ${\tau }_1$ for different values of the quantum efficiency $\eta$: from top to bottom $\eta =1.0$, $0.75$, and $0.5$. We set ${\tau }_2=1∕2$ and $g=g_{\mathrm{s}}$ (symmetric cloning). The dashed line is the value $2∕3$. The fidelity does not depend on the coherent state amplitude.

Figure 3

Plot of the fidelity $F_{\eta }\left(\xi \right)$ of the squeezed state $\mid \alpha ,\xi ⟩$ as a function of $\mid \xi \mid$ for different values of $\eta$: from top to bottom $\eta =1.0$, $0.75$, and $0.5$. We set ${\tau }_1={\tau }_2=1∕2$ and $g=g_{\mathrm{s}}$ (symmetric cloning). The dashed line is the value $2∕3$. The fidelity does not depend on the displacement amplitude $\alpha$.

Figure 4

Plot of the average fidelity ${\overline{F}}_{\eta }$ of a set of squeezed states as a function of ${\sigma }_{\mathrm{s}}$ (see text for details) and different values of the efficiency $\eta$: from top to bottom $\eta =1.0$, $0.75$, and $0.5$. The dashed line corresponds to $2∕3$; i.e., the optimal cloning fidelity of coherent states. We put ${\tau }_1={\tau }_2=1∕2$ and $g=g_{\mathrm{s}}$.

Figure 5

Plot of the input-output fidelity $F_{\eta }\left(N\right)$ of the displaced thermal state ${\varrho }_{\mathrm{th},\alpha }$ as a function of the average number of thermal photons $N$ and different values of the efficiency $\eta$: from top to bottom $\eta =1.0$, $0.75$, and $0.5$. The dashed line corresponds to $2∕3$; i.e., the optimal cloning of coherent states. We put ${\tau }_1={\tau }_2=1∕2$ and $g=g_{\mathrm{s}}$.

Figure 6

Plot of the average fidelity ${\overline{F}}_{\eta }$ of the set of thermal states distributed according to the top-hat distribution (53) as a function of the threshold value $\mathcal{N}$ and different values of the efficiency $\eta$: from top to bottom $\eta =1.0$, $0.75$, and $0.5$. The dashed line corresponds to $2∕3$. We put ${\tau }_1={\tau }_2=1∕2$ and $g=g_{\mathrm{s}}$.

Figure 7

Plot of the average fidelity ${\overline{F}}_{\eta }$ of the set of thermal states distributed according to a “half-Gaussian” distribution (54) as a function of ${\mu }_N$ and different values of the efficiency $\eta$: from top to bottom $\eta =1.0$, $0.75$, and $0.5$. The dashed line corresponds to $2∕3$. We put ${\tau }_1={\tau }_2=1∕2$ and $g=g_{\mathrm{s}}$.