Genuine multipartite entanglement is not a precondition for secure conference key agreement

Entanglement plays a crucial role in the security of quantum key distribution. A secret key can only be obtained by two parties if there exists a corresponding entanglement-based description of the protocol in which entanglement is witnessed, as shown by Curty et al. [M. Curty, M. Lewenstein, and N. Lütkenhaus, Phys. Rev. Lett. 92, 217903 (2004)]. Here we investigate the role of entanglement for the generalization of quantum key distribution to the multipartite scenario, namely, conference key agreement. In particular, we ask whether the strongest form of multipartite entanglement, namely, genuine multipartite entanglement, is necessary to establish a conference key. We show that, surprisingly, a nonzero conference key can be obtained even if the parties share biseparable states in each round of the protocol. Moreover, we relate conference key agreement with entanglement witnesses and show that a nonzero conference key can be interpreted as a nonlinear entanglement witness that detects a class of states which cannot be detected by usual linear entanglement witnesses.

Figure 1

Sketch of the considered CKA scenario: At each round of the protocol, a quantum resource is distributed to the parties (in our particular case, a multipartite quantum state). The parties perform local measurements and utilize the outcomes to extract a common secret key.

Figure 2

Asymptotic secret key rate, $r_{\mathrm{N}-\mathrm{BB}84}^{\infty }(N,k)$, for the family of states ${\rho }_{A{B}_1,\cdots ,B_{N-1}}^{(N,k)}$, Eq. (7), for the $N$-BB84 CKA protocol. The curves (straight lines) represent different values of $k$ (light grey: from left to right, $k=2, k=5, k=10$; dark grey: top, $k=N-1$, bottom, $k=\frac{N}{2}$). We also plot the key rate of CKA based on multiple noiseless bipartite QKD protocols (dashed line), both as a function of $N$. We remark that since $k\le N-1$, the curves for fixed $k$ start at different values of $N$.

Figure 3

Plot of the asymptotic key rate of the N-BB84 protocol for the state of Eq. (7) undergoing local depolarizing noise (solid lines), as a function of the depolarizing channel parameter $p$, for fixed $N=6$ and different $k$: $k=4$ (blue, left) and $k=5$ (green, right). The results are compared with the key rate of a concatenation of noisy bipartite BB84 QKD protocols (red dashed line).

Figure 4

Schematic representation of the set of tripartite states, adapted from Ref. [38]. In light blue (outer set) is represented the set of GME states. In red (dark grey area) is highlighted the set of biseparable states that are not separable with respect to any fixed partition, whereas in yellow (light grey) are represented the sets of states that are separable with respect to a fixed partition. In the middle, in green, is represented the set of fully separable states. A linear witness defines a hyperplane in the space of states. A nonzero conference key rate can be seen as a nonlinear entanglement witness, as it can detect states in the red area, i.e., outside a nonconvex set.

Figure 5

Left panel: Plot of the asymptotic key rate of the N-BB84 protocol for the state of Eq. (C2) (solid lines) as a function of $p$, for fixed $N=6$ and different $k$: $k=4$ (blue, left) and $k=5$ (green, right). The results are compared with the key rate of a concatenation of BB84 QKD protocols, given in Eq. (C6) (red dashed line), for $N=6$, as a function of $p$. Right panel: Plot of the asymptotic key rate of the N-BB84 protocol for the state of Eq. (C2) (solid lines) as a function of $p$, for a fixed value of $N=13$ and different values of $k$: $k=7$ (blue, left), $k=9$ (green, middle) and $k=11$ (black, right). The results are compared with the key rate of a concatenation of BB84 QKD protocols, given in Eq. (C6) (red dashed line), for $N=13$, as a function of $p$.