Quantum Teleportation Between Discrete and Continuous Encodings of an Optical Qubit

The transfer of quantum information between physical systems of a different nature is a central matter in quantum technologies. Particularly challenging is the transfer between discrete and continuous degrees of freedom of various harmonic oscillator systems. Here we implement a protocol for teleporting a continuous-variable optical qubit, encoded by means of low-amplitude coherent states, onto a discrete-variable, single-rail qubit—a superposition of the vacuum and single-photon optical states—via a hybrid entangled resource. We test our protocol on a one-dimensional manifold of the input qubit space and demonstrate the mapping onto the equator of the teleported qubit’s Bloch sphere with an average fidelity of $0.83\pm 0.04$. Our work opens up the way to the wide application of quantum information processing techniques where discrete- and continuous-variable encodings are combined within the same optical circuit.

Figure 1

Conceptual scheme of the experiment. The teleportation occurs from a CV qubit (a coherent state in our case) in input mode $A$ onto the DV vacuum-photon superposition in mode $D$. The preparation of the hybrid entangled resource (1) in modes $C$ and $D$ is heralded by a count of ${\mathrm{SPCM}}_1$, while the Bell state projection is conditioned on a click of ${\mathrm{SPCM}}_2$. The teleported state is measured by Bob using a homodyne detector. Another homodyne detector in mode $C$ serves the auxiliary purpose of the phase measurement.

Figure 2

Wigner functions of the CV mode of the entangled resource state, reconstructed with a 54% efficiency correction, projected in the DV mode onto various qubit states: (a) ${|1⟩}_D$; (b) ${|0⟩}_D$; (c) $|+⟩=({|0⟩}_D+{|1⟩}_D)/\sqrt{2}$; (d) $|-⟩=({|0⟩}_D-{|1⟩}_D)/\sqrt{2}$.

Figure 3

Experimental results. (a) Density matrices of the teleported state, as measured by Bob’s homodyne detector, for different phases of the input qubit, reconstructed with a correction for 54% of detection efficiency. Top row: Real part. Bottom row: Imaginary part. The coloring scheme distinguishes the diagonal (gray) as well as positive (purple) and negative (green) off-diagonal matrix elements for viewing convenience. (b) Off-diagonal elements of the teleported qubit’s density matrices, dots. Lines: Simulation results. Blue (green) data: With (without) efficiency correction. The radial lines indicate the corresponding input state phases. (c) Color map: Expected fidelity of the protocol, obtained via a numeric simulation, over the full Bloch sphere. Dots show the states tested in the experiment. White: Target. Black: Measured. (d) Fidelity between the states reconstructed from the experimental data with an efficiency correction and theoretical target states. Error bars correspond to one standard deviation. The solid lines are theoretical predictions taking into account all experimental imperfections including the phase estimation uncertainty. The red dashed line ignores the phase uncertainty.