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. 2017 Apr 26;8:15043. doi: 10.1038/ncomms15043

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Copyright © 2017, The Author(s)

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(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 Λ_i, 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 Λ_i into a single ‘extended' LOCC Δ_i while the resource state σ can be stretched back in time and out of the adaptive operations. We may therefore write =Δ_i(⊗σ). (d) We iterate the previous steps for all transmissions, so as to stretch n copies σ^⊗n and collapse all the extended LOCCs Δ_n o …o Δ₁ into a single LOCC Λ. In other words, we may write =Λ(⊗σ^⊗n). (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 =(σ^⊗n) for a trace-preserving LOCC . The procedure is asymptotic in the presence of asymptotic channel simulations (bosonic channels).

Inline graphic — (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 Λ_i, 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 Λ_i into a single ‘extended' LOCC Δ_i while the resource state σ can be stretched back in time and out of the adaptive operations. We may therefore write =Δ_i(⊗σ). (d) We iterate the previous steps for all transmissions, so as to stretch n copies σ^⊗n and collapse all the extended LOCCs Δ_n o …o Δ₁ into a single LOCC Λ. In other words, we may write =Λ(⊗σ^⊗n). (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 =(σ^⊗n) for a trace-preserving LOCC . The procedure is asymptotic in the presence of asymptotic channel simulations (bosonic channels).