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 \(\left|{\psi }_{{{{{{{{\rm{fe}}}}}}}}}\right\rangle\), 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 \(\left|{\psi }_{{{{{{{{\rm{in}}}}}}}}}\right\rangle\) 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.