Upper Bounds on Device-Independent Quantum Key Distribution Rates in Static and Dynamic Scenarios

In this work, we develop upper bounds on key rates for device-independent quantum key distribution (DI-QKD) protocols and devices. We study the reduced $\mathrm{cc}$-squashed entanglement and show that it is a convex functional. As a result, we show that the convex hull of the currently known bounds is a tighter upper bound on the device-independent key rates of the standard Clauser-Horne-Shimony-Holt (CHSH)-based protocol. We further provide tighter bounds for DI-QKD key rates achievable by any protocol applied to the CHSH-based device. This bound is based on reduced relative entropy of entanglement optimized over decompositions into local and nonlocal parts. In the dynamical scenario of quantum channels, we obtain upper bounds for device-independent private capacity for the CHSH-based protocols. We show that the device-independent private capacity for the CHSH-based protocols on depolarizing and erasure channels is limited by the secret key capacity of dephasing channels.

Figure 1

In this figure, we show the plots for standard device-independent CHSH protocol obtained in Refs. [41, 42], and the upper bound given in Theorem 2, which is the convex hull of the former bounds, depicted in green.

Figure 2

In this plot, we depict the bounds on the amount of DI key that can be obtained from a CHSH-based device. The yellow line and green line corresponds to the upper bounds obtained from Eq. (3). The blue line corresponds to the bound obtained in Appendix B of Ref. [41]. The purple line corresponds to the bound obtained in Ref. [38]. The red line corresponds to the lower bounds obtained in Ref. [60].

Figure 3

In the above figure, we plot upper bounds on the device-dependent QKD capacities of depolarizing channel (yellow line), dephasing channel (blue line), and erasure channel (green line). We notice that the upper bounds for erasure and dephasing channels are achievable device-dependent QKD rates (capacities). We then notice that for the CHSH protocols, the upper bounds on the DI-QKD capacities of channels is limited by the device-dependent QKD capacity of dephasing channels.