Upper bounds on secret-key agreement over lossy thermal bosonic channels

Upper bounds on the secret-key-agreement capacity of a quantum channel serve as a way to assess the performance of practical quantum-key-distribution protocols conducted over that channel. In particular, if a protocol employs a quantum repeater, achieving secret-key rates exceeding these upper bounds is evidence of having a working quantum repeater. In this paper, we extend a recent advance [Liuzzo-Scorpo et al., Phys. Rev. Lett. 119, 120503 (2017)] in the theory of the teleportation simulation of single-mode phase-insensitive Gaussian channels such that it now applies to the relative entropy of entanglement measure. As a consequence of this extension, we find tighter upper bounds on the nonasymptotic secret-key-agreement capacity of the lossy thermal bosonic channel than were previously known. The lossy thermal bosonic channel serves as a more realistic model of communication than the pure-loss bosonic channel, because it can model the effects of eavesdropper tampering and imperfect detectors. An implication of our result is that the previously known upper bounds on the secret-key-agreement capacity of the thermal channel are too pessimistic for the practical finite-size regime in which the channel is used a finite number of times, and so it should now be somewhat easier to witness a working quantum repeater when using secret-key-agreement capacity upper bounds as a benchmark.

Figure 1

The figures plot upper bounds on the nonasymptotic secret-key-agreement capacity of the thermal channels of transmissivity $\eta \in (0,1)$ and thermal mean photon number $N_B>0$ given by the second-order approximation from (9). In each figure, we select certain values of $\eta$ (corresponding to a certain distance $L$ via $\eta =exp[-L/L_0]$) and $N_B$, as indicated below each figure. In all cases, we take the conservative value of $\varepsilon ={10}^{-10}$ as indicated in Ref. [67]. Each figure indicates that the asymptotic secret-key-agreement capacity is too pessimistic of a benchmark for demonstrating a quantum repeater when using the channel a finite number of times. That is, there is an appreciable difference between the asymptotic and non-asymptotic secret-key-agreement capacity. The parameter values that we take in each figure are as follows: (a) $N_B=0.1$, $L=26.49$ km, $\eta =0.3$, (b) $N_B=0.1$, $L=11.23$ km, $\eta =0.6$, (c) $N_B={10}^{-3},$ $L=50.65$ km, $\eta =0.1$, (d) $N_B={10}^{-3}$, $L=15.25$ km, $\eta =0.5$, (e) $N_B=3\times {10}^{-7}$, $L=26.49$ km, $\eta =0.3$, and (f) $N_B=3\times {10}^{-7}$, $L=15.25$ km, $\eta =0.5$.