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 \(|{\psi }_{{{{{{{{\rm{hg}}}}}}}}}\rangle\) 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 \(|{\psi }_{{{{{{{{\rm{fe}}}}}}}}}\rangle\) and \(|{\psi }_{{{{{{{{\rm{hg}}}}}}}}}\rangle\) 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 \(|{\psi }_{{{{{{{{\rm{av}}}}}}}}}\rangle\), with \(\eta =\sin (2\theta )\). 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 \(|{\psi }_{{{{{{{{\rm{fe}}}}}}}}}\rangle\). 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.