Relativistic Quantum Teleportation with Superconducting Circuits

We study the effects of relativistic motion on quantum teleportation and propose a realizable experiment where our results can be tested. We compute bounds on the optimal fidelity of teleportation when one of the observers undergoes nonuniform motion for a finite time. The upper bound to the optimal fidelity is degraded due to the observer’s motion. However, we discuss how this degradation can be corrected. These effects are observable for experimental parameters that are within reach of cutting-edge superconducting technology.

Figure 1

Alice and Rob initially share a two-mode squeezed cavity state that is produced by the EPR source. Consecutively, Rob’s cavity undergoes nonuniform motion consisting of segments of constant acceleration (green hyperbolae) and inertial coasting (red parallel lines). Alice, who remains inertial, sends the outcome of the Bell measurement on the input state and her mode of the entangled state to Rob via a classical channel (blue wavy arrow). Rob can then retrieve the teleported state by performing the appropriate unitary $U$. In addition to the standard protocol, both Alice and Rob measure their respective proper times and perform local rotations to compensate for the phases accumulated during the motion.

Figure 2

Sketch of the experimental setup. A coplanar waveguide is interrupted by two superconducting quantum interference devices (SQUIDs) placed at a fixed distance $L_0$, creating a cavity of effective length $L_{\mathrm{eff}}=L_0+d_{+}(t)+d_{-}(t)$. The time dependence of the distances $d_{\pm }(t)$ between the SQUIDs and the effective boundaries is controlled with external drive fields applied to the superconducting circuits to simulate a cavity of constant length with respect to its rest frame.

Figure 3

The fidelities $\tilde{\mathcal{F}}$ and ${\tilde{\mathcal{F}}}_{\mathrm{opt}}$ are plotted in (a) and (b), respectively, as functions of Rob’s proper time $\tau$ and the acceleration $a$. The plots are shown for modes $k=1$ (Alice) and $k^{\prime }=3$ (Rob). For the cavity length we use a typical value of $L=1.2\mathrm{cm}$. Together with the speed of light $c=1.2\times {10}^8\mathrm{m}/\mathrm{s}$, we obtain a fundamental frequency ${\omega }_k=2\pi \times c/(2L)=2\pi \times 5\mathrm{GHz}$ and ${\omega }_{k^{\prime }}=3{\omega }_k$. The squeezing parameter is $r=1/2$ and the maximum value of the perturbative parameter is $h^2=0.06$. Once the effect of the free time evolution in (a) is removed, the correction due to acceleration can be isolated in (b).