Gain tuning for continuous-variable quantum teleportation of discrete-variable states

We present a general formalism to describe continuous-variable (CV) quantum teleportation of discrete-variable (DV) states with gain tuning, taking into account experimental imperfections. Here the teleportation output is given by independently transforming each density matrix element of the initial state. This formalism allows us to accurately model various teleportation experiments and to analyze the gain dependence of their respective figures of merit. We apply our formalism to the recent experiment on CV teleportation of qubits [S. Takeda et al., Nature 500, 315 (2013)] and investigate the optimal gain for the transfer fidelity. We also propose and model an experiment for CV teleportation of DV entanglement. It is shown that, provided the experimental losses are within a certain range, DV entanglement can be teleported for any nonzero squeezing by optimally tuning the gain.

Figure 1

Schematic of CV teleportation. HD, homodyne detection.

Figure 2

Simulation results of CV teleportation of a dual-rail qubit. (a) $g$ dependence of $F_{\mathrm{state}}$ at $r=1.0$. (b) $r$ dependence of $g_{\mathrm{state}}^{\mathrm{opt}}$. (c) $r$ dependence of $F_{\mathrm{state}}^{\max }$. (d) $g$ dependence of $F_{\mathrm{qubit}}$ at $r=1.0$. (e) $r$ dependence of $g_{\mathrm{qubit}}^{\mathrm{opt}}$ at $\eta =0.7$. (f) $r$ dependence of $F_{\mathrm{qubit}}^{\max }$ and $P_{\mathrm{qubit}}$ at $\eta =0.7$. (g) Experimental ${\widehat{\rho }}_{\mathrm{out}}$ in Ref. [30]. (h) Theoretical ${\widehat{\rho }}_{\mathrm{out}}$ in Eq. (31) for $(\alpha ,\beta ,{\eta }_1,{\eta }_2,r,l,g)=(1/\sqrt{2},-i/\sqrt{2},0.69,0.06,1.01,0.2,0.79)$.

Figure 3

Schematic of hybrid entanglement swapping.

Figure 4

Simulation results of entanglement swapping. (a) Area of $(r,g)$ where PPT of ${\widehat{\rho }}_{\mathrm{XZ}}$ is violated is shaded (green) [Eq. (44)]. (b) $r$ dependence of $g_{\mathrm{LN}}^{\mathrm{opt}}$. (c) $r$ dependence of $E_{\mathrm{LN}}^{\mathrm{opt}}\left({\widehat{\rho }}_{\mathrm{XZ}}\right)$. The LN of the initial state is $E_{\mathrm{LN}}^{\max }\left({\widehat{\rho }}_{\mathrm{XY}}\right)=1$ for $\eta =1.0$ and $E_{\mathrm{LN}}^{\max }\left({\widehat{\rho }}_{\mathrm{XY}}\right)=0.548$ for $0.7$. (d) Theoretical ${\widehat{\rho }}_{\mathrm{XY}}$ in Eq. (38) for $\eta =0.8$ (real part). (e) Theoretical ${\widehat{\rho }}_{\mathrm{XZ}}$ in Eq. (39) for $(\eta ,r,l,g)=(0.8,1.0,0.2,0.90)$ (real part). Note that there is no imaginary component for either ${\widehat{\rho }}_{\mathrm{XY}}$ or ${\widehat{\rho }}_{\mathrm{XZ}}$.