Figure 7 : Two-way capacities for Gaussian channels in terms of the relevant channel parameters.

From: Fundamental limits of repeaterless quantum communications

Figure 7

(a) Two-way capacity of the thermal-loss channel as a function of transmissivity η for =1 thermal photon. It is contained in the shadowed area identified by the lower bound (LB) and upper bound (UB) of equation (25). Our upper bound is clearly tighter than those based on the squashed entanglement, computed in ref. 18 (dotted) and ref. 54 (dashed). Note that at high transmissivities. For =0 (lossy channel) the shadowed region shrinks into a single line. (b) Two-way capacity of the amplifier channel as a function of the gain g for =1 thermal photon. It is contained in the shadowed specified by the bounds in equation (27). For small gains, we have . For =0 (quantum-limited amplifier) the shadowed region shrinks into a single line. (c) Two-way capacity of the additive-noise Gaussian channel with added noise ξ. It is contained in the shadowed region specified by the bounds in equation (30). For small noise, we have . Our upper bound is much tighter than those of ref. 18 (dotted), ref. 54 (dashed) and ref. 51 (dot-dashed).