Effect of relativistic acceleration on continuous variable quantum teleportation and dense coding

We investigate how relativistic acceleration of the observers can affect the performance of the quantum teleportation and dense coding for continuous variable states of localized wave packets. Such protocols are typically optimized for symmetric resources prepared in an inertial frame of reference. A mismatch of the sender’s and the receiver’s accelerations can introduce asymmetry to the shared entanglement, which has an effect on the efficiency of the protocol that goes beyond entanglement degradation due to acceleration. We show how these asymmetry losses can be reduced by an extra local operations and classical communication (LOCC) step in the protocols.

Figure 1

The trajectories of accelerating parties Alice (right) and Bob (left) who perform a quantum-information protocol using the state of inertial wave packets ${\varphi }_{\mathrm{I}}$ and ${\varphi }_{\mathrm{II}}$ as a resource. The two Rindler wedges intersect to allow for classical communication. The wave packets are drawn outside the intersection region for clarity.

Figure 2

The initial condition for the inertial wave packet ${\varphi }_{\mathrm{I}}$ and the accelerating wave packet ${\psi }_{\mathrm{I}}$. We obtain ${\psi }_{\mathrm{I}}$ by deforming ${\varphi }_{\mathrm{I}}$ in the same way in which the modes of a Dirichlet cavity deform under acceleration. The initial conditions for ${\varphi }_{\mathrm{II}}$ and ${\psi }_{\mathrm{II}}$ are the same as shown here but mirrored with respect to spacetime origin.

Figure 3

The dependence of the overlap ${\alpha }_{\mathrm{\Lambda }}=({\psi }_{\mathrm{\Lambda }},{\varphi }_{\mathrm{\Lambda }})$, $\mathrm{\Lambda }\in \{\mathrm{I},\mathrm{II}\}$, on the proper acceleration of the wave packet ${\psi }_{\mathrm{\Lambda }}$. Since we put $x_0=1/\mathcal{A}$ in Eqs. (11) and (13), for $\mathcal{A}\to 0$, the initial conditions escape to infinity. We remedy this by interpolating between $\mathcal{A}=0$ and $\mathcal{A}=0.03$.

Figure 4

The logarithmic negativity $\mathcal{E}(\mathcal{A})$ of a resource state as detected by observers Alice and Bob (corresponding to wave packets ${\psi }_{\mathrm{I}}$ and ${\psi }_{\mathrm{II}}$) moving with equal-magnitude accelerations ${\mathcal{A}}_{\mathrm{I}}={\mathcal{A}}_{\mathrm{II}}=\mathcal{A}$. The original state is a two-mode squeezed vacuum state characterized by a squeezing coefficient $r$ of the inertial wave packets ${\varphi }_{\mathrm{I}}$ and ${\varphi }_{\mathrm{II}}$.

Figure 5

The logarithmic negativity $\mathcal{E}(\mathcal{A})$ of a two-mode squeezed vacuum resource state as detected by asymmetrically accelerating Alice and Bob. The original state is characterized by a squeezing coefficient $r=3.0$.

Figure 6

The fidelity $\mathcal{F}$ of quantum teleportation performed by Alice and Bob moving with equal-magnitude accelerations ${\mathcal{A}}_{\mathrm{I}}={\mathcal{A}}_{\mathrm{II}}=\mathcal{A}$. The entangled state used as a resource is a two-mode squeezed vacuum state of ${\varphi }_{\mathrm{I}}$ and ${\varphi }_{\mathrm{II}}$. $r$ is the squeezing coefficient characterizing the resource.

Figure 7

The teleportation fidelity $\mathcal{F}$ in the case when the magnitudes of Alice’s and Bob’s accelerations are independent. Fidelity values below 0.5 are clipped as $\mathcal{F}=0.5$ is achievable with a classical strategy. $r$ is the squeezing coefficient of the two-mode squeezed vacuum resource.

Figure 8

The teleportation fidelity of the protocol we consider (yellow, checked) and the lower bound on optimal teleportation fidelity (blue) as a function of Alice’s and Bob’s accelerations, ${\mathcal{A}}_{\mathrm{I}}$ and ${\mathcal{A}}_{\mathrm{II}}$. The resource state is characterized by the squeezing parameter $r=3.5$.

Figure 9

The mutual information $H$ of a dense coding protocol performed by Alice and Bob moving with equal-magnitude accelerations ${\mathcal{A}}_{\mathrm{I}}={\mathcal{A}}_{\mathrm{II}}=\mathcal{A}$. $r$ is the squeezing coefficient of the resource state.

Figure 10

The mutual information $H$ for independent magnitudes of Alice’s and Bob’s accelerations. $r$ is the squeezing coefficient of the resource state.

Figure 11

(a) The teleportation fidelity without (yellow, checked) and with (red) the asymmetry-compensating LOCC. (b) Mutual information of the dense coding protocol without (yellow, checked) and with (red) asymmetry compensation. In both cases, the losses due to unequal accelerations of the observers are almost completely removed. The resource state is characterized by the squeezing parameter $r=3.5$.