Device-independent conference key agreement from two-body correlators with balanced Dicke states

Abstract

Device-independent conference key agreement (DI-CKA) enables $N > 2$ distant parties to establish a common secret key without detailed characterization of the source or measurement devices. Many existing protocols rely on full-body Bell correlators, whose measurement can be demanding for a large number of parties. Here, we show that two-body correlators alone suffice for DI-CKA using balanced Dicke states for every even $N \geq 4$. We construct two-body Bell inequalities and derive a tight analytical lower bound on the conditional von Neumann entropy by comparing their Bell operators with those of asymmetric Clauser-Horne-Shimony-Holt inequalities. In the ideal setting, certified key rates can approach $1-h[(N-2)/(4(N-1))]$, where $h$ is the binary entropy function. This rate remains bounded away from zero as $N \rightarrow \infty$, tending to $1-h(1/4) \approx 0.189$ bits per key-generation round. We show that the honest implementation of our protocol is invariant under collective phase noise. Moreover, a complementary full-body Bell construction certifies positive conference-key rates from correlations arbitrarily close to the local set.

Can balanced Dicke states support device-independent conference key agreement (DI-CKA) using only two-body Bell correlators, despite imperfect pairwise key correlations? The paper constructs such protocols for every even $N\geq4$, with asymptotic security analyzed under an i.i.d. assumption. Alice uses two binary measurements and each Bob three; all parties measure $X$ for key generation. A parameterized family of two-body Bell inequalities has local bound $\max\{2,2\alpha\}$ and quantum maximum $2\sqrt{1+\alpha^2}$. Grouping the Bobs into a composite party and defining sign observables for sums of their measurements yields an operator comparison with asymmetric CHSH. This transfers a tight analytical conditional-entropy bound, including noisy preprocessing; tightness follows from embedding a saturating bipartite strategy. For balanced Dicke states, suitable measurements make the certified entropy approach one bit as $\alpha\to0$. The resulting ideal key rate approaches $1-h[(N-2)/(4(N-1))]$ bits per key-generation round, tending to $1-h(1/4)\approx0.189$ as $N\to\infty$. A separate full-body construction certifies the same rate for correlations arbitrarily close to the local set: their distance is bounded by $(N-1)\epsilon/2$, while maximal Bell violation certifies one bit of conditional entropy for every $\epsilon>0$. The honest Dicke-state implementation is invariant under collective phase noise. However, both Dicke protocols tolerate less local depolarizing noise and detection inefficiency than the compared GHZ protocol. The two-body construction is particularly sensitive because high entropy requires small $\alpha$, shrinking the Bell-violation margin. The reported rates are ideal or noise-modeled asymptotic rates, not finite-key guarantees.