Quantum channel correction outperforming direct transmission

Abstract

Long-distance optical quantum channels are necessarily lossy, leading to errors in transmitted quantum information, entanglement degradation and, ultimately, poor protocol performance. Quantum states carrying information in the channel can be probabilistically amplified to compensate for loss, but are destroyed when amplification fails. Quantum correction of the channel itself is therefore required, but break-even performance-where arbitrary states can be better transmitted through a corrected channel than an uncorrected one-has so far remained out of reach. Here we perform distillation by heralded amplification to improve a noisy entanglement channel. We subsequently employ entanglement swapping to demonstrate that arbitrary quantum information transmission is unconditionally improved-i.e., without relying on postselection or post-processing of data-compared to the uncorrected channel. In this way, it represents realization of a genuine quantum relay. Our channel correction for single-mode quantum states will find use in quantum repeater, communication and metrology applications.

Fig. 1. Conceptual representations of quantum state transmission through a lossy channel, with and without correction.

a A quantum state, here a qubit encoded in a single mode “e'', is transmitted through a lossy channel, which degrades the state quality. b After the loss, the noisy state can be corrected using a heralded amplifier (HA). Mode “a” carries the ancilla photon that powers the HA. The operation of HA has an independent success signal, so postselection is not required. However, HA failure destroys the state. c By adding a mode-entangled state $∣{\psi }_{\mathrm{fe}}⟩$, success of the HA heralds a noise-corrected quantum channel. This can be used upon success by teleporting a qubit in mode “g” onto mode “v” via a Bell state measurement (BSM) between “g” and “f''. d Instead of transmitting a qubit $∣{\psi }_{\mathrm{in}}⟩$ through the lossy or corrected channel, it is possible to transmit half of an entangled state, leading to distributed entanglement through a heralded corrected channel in the last case.

Fig. 2. Concept and layout of the experimental apparatus.

a Concept of the experiment, which implements the scheme of Fig. 1d. Half of the entangled state $\mid {\psi }_{\mathrm{hg}}⟩$ is teleported through a heralded corrected channel. This comprises an entangled source, loss on one mode (the distribution mode), and an HA. This leads to entanglement between modes “h” and “v” that, ideally, is larger than the entanglement between “f” and “e” after “e” goes through the lossy channel. Modes “h” and “v” are then brought together for a joint measurement to determine the concurrence C. b Experimental setup for the error-corrected quantum channel. This diagram has a one-to-one mode mapping to part a, as shown in subfigure c, but some of the modes are now spatially overlaid and distinguished by polarization. Blue and green backgrounds highlight ES and HA stages, respectively. Gray background highlights polarization dependent loss. Optical axes of HWP_fe and HWP_hg, set at π/8 with respect to the horizontal axis, prepared $\mid {\psi }_{\mathrm{fe}}⟩$ and $\mid {\psi }_{\mathrm{hg}}⟩$ states, respectively. Optical modes input to PBS_HA2 and PBS_ES2 were mixed by HWP_HA and HWP_ES, set at π/8. HWP_η(θ), with optic axis orientated at angle θ, initialized the HA resource state $\mid {\psi }_{\mathrm{av}}⟩$, with $\eta =\sin \left(2\theta \right)$. HA success was heralded by a single detection event in either superconducting nanowire single photon detector (SNSPD) D₁ or D₂ and ES success was heralded by a single detection event in either D₃ or D₄. Detectors D₊ and D₋, together with QWP_o, HWP_o and PBS_o were used to perform polarization state tomography on the modes “h” and “v''. c Mode propagation inside the setup in case of the error-corrected channel. Mode “h” propagates through the setup unaffected by loss or other optical components. d Same setup but with single photon input, used to test direct transmission through loss to compare the final concurrences. HWP_p was set to π/8 in order to prepare $\mid {\psi }_{\mathrm{fe}}⟩$. QWP_d, HWP_d and PBS_HA1, together with detectors D₊ and D₁ were used to perform polarization tomography of the state after the loss. Note that the ultrahigh-heralding-efficiency photon sources required for this protocol are not shown in the figure---see Methods for details.

Fig. 3. Concurrence measurements of entanglement distributed over different types of channels for three different values of added loss L on mode “e”.

Orange lines and blue shaded areas correspond to the experimentally measured and theoretically predicted concurrence C_fe of $∣{\psi }_{\mathrm{fe}}⟩$ state distributed directly through loss. Red dots and gray shaded areas correspond to the experimentally measured and theoretically predicted concurrence C_hv of entanglement distributed via the error-corrected quantum channel Fig. 1d, with the same amount of loss and the same input state $∣{\psi }_{\mathrm{fe}}⟩$. Black triangles highlight data points with maximum observed increase in concurrence. The upper and lower bounds for the theoretical predictions correspond to the lowest and highest observed transmissions for each mode inside the experimental setup, respectively. Error bars and shaded areas on orange lines correspond to the experimentally observed statistical uncertainty of ± 1 standard deviation.

Fig. 4. Absolute values of the density matrix elements for quantum states distributed via different types of channels.

a Maximally-mode-entangled state sent through an uncorrected quantum channel with L = 0.988 ± 0.001 added loss. b Same input state, but distributed through the error-corrected quantum channel with the nominal gain setting from Fig. 3, at which the observed concurrence was the highest. c, d and e, f same as a, b, but with L = 0.958 ± 0.001 and L = 0.903 ± 0.002, respectively.