Versatility of continuous-variable asymmetric tripartite entanglement allows Alice and Clare to keep secrets from Bob

The fully symmetric Gaussian tripartite entangled pure states will not exhibit two-mode Einstein-Podolsky-Rosen (EPR) steering. This means that any two participants cannot share quantum secrets using the security of one-sided device independent quantum key distribution (1SDI-QKD) without involving the third. They are restricted at most to standard quantum key distribution, which is less secure. Here we demonstrate an asymmetric tripartite system that can exhibit bipartite EPR steering, so that two of the participants can use 1SDI-QKD without involving the other. This is possible because the promiscuity relations of continuous-variable tripartite entanglement are different from those of discrete-variable systems. We analyze these properties for two different systems, showing that the asymmetric system exhibits practical properties not found in the symmetric one.

Figure 1

The beam-splitter model used for symmetric systems. Three optical parametric oscillators produce outputs squeezed in the $\widehat{Y}$ (${\mathrm{OPO}}_1$) or $\widehat{X}$ (${\mathrm{OPO}}_2$ and ${\mathrm{OPO}}_3$) quadratures. The outputs are combined with two beam splitters, producing three output fields ${\widehat{b}}_1,{\widehat{b}}_2,{\widehat{b}}_3$.

Figure 2

$D{S}_{13}^{-}, V_{123}, V_{312}$, and $V_{13}$ for the asymmetric model, with ${\kappa }_2=0.6{\kappa }_1$. We see that both bipartite and tripartite entanglement are predicted over a range of interaction strength $\zeta t$. Note that the line at 4 is a guide to the eye.

Figure 3

The EPR-steering correlations for modes 1 and 3 of the asymmetric model, with ${\kappa }_2=0.6{\kappa }_1$. Note that the line at 1 is a guide to the eye.

Figure 4

The minimum bit rates for modes 1 and 3 of the asymmetric model, with ${\kappa }_2=0.6{\kappa }_1$, and the minimum rate for any pair of the symmetric scheme. $K_{ij}$ is the minimum bit rate for mode $j$ sending a quantum key to mode $i$. A positive value shows that 1SDI-QKD is possible. The line at zero is a guide to the eye.

Figure 5

The output spectral EPR-steering correlations for $\epsilon =0.8{\epsilon }_c$. We see that Bob can now be steered by Alice, although the violation of the inequality is much less than those for Alice and Clare. $\omega$ represents the frequency in terms of ${\gamma }_1$ around the cavity resonance.

Figure 6

The spectral minimum bit rates for Alice sending a quantum key to Bob, and Alice and Clare sending to each other. A positive value shows that 1SDI-QKD is possible. Note that the frequency axis is narrower than for Fig. 5, reflecting the fact that the bandwidth for 1SDI-QKD is less than that for EPR steering. The line at zero is a guide to the eye.

Figure 7

The minimum bit rates for modes 1 and 3 of the asymmetric model, with ${\kappa }_2=0.6{\kappa }_1$, and the minimum rate for any pair of the symmetric scheme. $K_{ij}$ is the minimum bit rate for mode $j$ sending a quantum key to mode $i$. A positive value shows that 1SDI-QKD is possible. The line at zero is a guide to the eye.