Fundamental limitations on the device-independent quantum conference key agreement

We provide several general upper bounds on the rate of a key secure against a quantum adversary in the device-independent conference key agreement (DI-CKA) scenario. They include bounds by reduced entanglement measures and those based on multipartite secrecy monotones such as a multipartite squashed entanglement-based measure, which we refer to as reduced c-squashed entanglement. We compare the latter bound with the known lower bound for the protocol of conference key distillation based on the parity Clauser-Horne-Shimony-Holt game. We also show that the gap between the DI-CKA rate and the device-dependent rate is inherited from the bipartite gap between device-independent and device-dependent key rates, giving examples that exhibit the strict gap.

Figure 1

Depiction of a construction of a tripartite state ${\rho }_{A\left(B_1{B}_2\right)C}$ with a gap $K_{\text{DI}}\left({\rho }_{A\left(B_1{B}_2\right)C}\right)<K_{\text{DD}}\left({\rho }_{A\left(B_1{B}_2\right)C}\right)$ from a state ${\rho }_{A{B}_1}$ (blue dotted line) satisfying $K^{\downarrow }\left({\rho }_{A{B}_1}\right)<K_{\text{DD}}\left({\rho }_{A{B}_1}\right)$ (provided in Ref. [16]) and any state ${\rho }_{B_{2}C}$ (green solid line) satisfying $K_{\text{DD}}\left({\rho }_{B_{2}C}\right)\ge K_{\text{DD}}\left({\rho }_{A{B}_1}\right)$, e.g., the singlet state.

Figure 2

Plot of upper and lower bounds on the DI-CKA of Ref. [20]. The yellow dashed line represents an upper bound (not optimized) on the upper bound $\frac{1}{N-1}I(N\left(A\right)\downarrow E)$ from Eq. (40) with the attack strategy in Eq. (68). The green dotted line represents an upper bound (not optimized) on $D_N(N\left(A\right)\downarrow E)$ from Corollary 3. The red dash-dotted curve is the trivial upper bound obtained in Corollary 6 via the relative entropy of entanglement bound ($1-\nu$). The blue solid line represents the lower bound from Ref. [20].