Optimal continuous-variable teleportation under energy constraint

Quantum teleportation is one of the crucial protocols in quantum information processing. It is important to accomplish an efficient teleportation under practical conditions, aiming at a higher fidelity desirably using fewer resources. The continuous-variable (CV) version of quantum teleportation was first proposed using a Gaussian state as a quantum resource, while other attempts were also made to improve performance by applying non-Gaussian operations. We investigate the CV teleportation to find its ultimate fidelity under energy constraint identifying an optimal quantum state. For this purpose, we present a formalism to evaluate teleportation fidelity as an expectation value of an operator. Using this formalism, we prove that the optimal state must be a form of photon-number entangled states. We further show that Gaussian states are near optimal, while non-Gaussian states make a slight improvement and therefore are rigorously optimal, particularly in the low-energy regime.

Figure 1

Minimum eigenvalues of the matrices with elements given by $f_{j,l}^{\left(0\right)}-f_{j,l}^{\left(d\right)}$. All eigenvalues lie in the positive region.

Figure 2

(a) Optimal fidelity achieved by PNESs (blue circles) compared to the fidelity by TMSVs (red curve) and by $|{\mathrm{\Psi }}^{\left(d\right)}\rangle$ with $d=1$ (brown triangles) and $d=2$ (purple squares) against energy $\langle {\widehat{n}}_A\rangle$. We also plot the fidelity attained by other non-Gaussian states: TMCs (green dot-dashed curve) and PSSVs (yellow dashed curve). The gray dotted line represents the no-cloning bound $F_{\mathrm{nc}}\approx 0.6826$. (b) Difference in the optimal fidelity between non-Gaussian states and Gaussian states against energy $\overline{n}$.

Figure 3

Photon-number distribution $P_j={\left|c_j\right|}^2$ of the optimal PNESs (blue circles) and of TMSVs (red squares), with the same mean photon number $\overline{n}\simeq 0.052$. Also the photon-number distributions of TMCs (green triangles) and PSSVs (yellow pluses) are shown.

Figure 4

Fidelity difference between Gaussian states and non-Gaussian states under the channel loss of transmittance $\eta$. The initial mean photon number $\overline{n}$ is chosen as 0.052 (blue solid curve), 0.204 (green dashed curve), and 0.515 (red dotted curve). The gray region represents that Gaussian states beat non-Gaussian states.