Figure 4 : Teleportation stretching of an adaptive quantum protocol.

From: Fundamental limits of repeaterless quantum communications

Figure 4

(a) Consider the ith transmission through channel , where the input (i−1)th register state is given by . After transmission through and the adaptive LOCC Λi, the register state is updated to . (b) Let us simulate the channel by a LOCC and a resource state σ. (c) The simulation LOCC can be combined with the adaptive LOCC Λi into a single ‘extended’ LOCC Δi while the resource state σ can be stretched back in time and out of the adaptive operations. We may therefore write i(σ). (d) We iterate the previous steps for all transmissions, so as to stretch n copies σn and collapse all the extended LOCCs Δn o …o Δ1 into a single LOCC Λ. In other words, we may write =Λ(σn). (e) Finally, we include the preparation of the separable state into Λ and we also average over all local measurements present in Λ, so that we may write the output state as =(σn) for a trace-preserving LOCC . The procedure is asymptotic in the presence of asymptotic channel simulations (bosonic channels).