Routed Bell tests with arbitrarily many local parties

TL;DR

This work addresses device-independent quantum key distribution (DIQKD) over long distances, where the detection-efficiency loophole impedes secure key generation. It introduces routed Bell tests with arbitrarily many local self-tests for both Alice and Bob, linked by entanglement swapping, and derives a BB84-type DI key-rate bound that depends continuously on the local self-test winning probability, via robust self-testing: $r_{\text{key}} \geq 1 - h(Q_Z) - h(Q_X) - \mathcal{O}(\sqrt{\varepsilon})$ as the self-test error $\varepsilon$ vanishes. A polynomial optimization framework based on the NPA hierarchy, augmented with reliable-estimates relaxations and commutator/anticommutator bounds, enables estimation of $H(A|E)$ under marginal constraints and full statistics. Numerical results show that employing two local switches yields advantages over a single-switch setup, and in the perfect self-test limit the asymptotic rate recovers the BB84 bound, highlighting practical potential for long-distance DIQKD with robust security guarantees.

Abstract

Device-independent quantum key distribution (DIQKD) promises cryptographic security based solely on observed quantum correlations, yet its implementation over long distances remains limited by the detection-efficiency loophole. Routed Bell tests have recently re-emerged as a promising strategy to mitigate this limitation by enabling local self-testing of one party's device. However, extending this idea to self-testing both communicating parties has remained unclear. Here, we introduce a modified setup that enables local self-tests for both Alice and Bob and analyze its security against potential attacks. Employing modern tools from robust self-testing, we show that in a BB84-type protocol between the self-tested devices, the achievable key rate varies continuously with the winning probability of the local tests. In particular, we find that perfect local Bell tests can, in principle, overcome the detection-efficiency barrier, rendering the asymptotic key rate limited only by standard bit-flip errors, as in the device-dependent case.

Figures (2)

• Figure 1: (a) depicts the routed Bell experiment proposed in previous work Tan_2024Lobo2024Le_Roy_Deloison_2025. (b) depicts our new proposal. This combines entanglement swapping, denoted as $S$, with local Bell tests, thereby enabling arbitrarily many local parties and associated random variables --- $T_A$ depicted with the red-lined options and $T_B$ depicted with the green-lined options. These random variables determine whether the bipartite state is sent to another local party or to the entanglement-swapping apparatus to produce the key-generating state.
• Figure 2: In the figure, we present numerical results obtained by solving \ref{['eq:optimization']} under a relaxation that replaces the marginal constraints with bounds on the commutator and anti-commutator derived from \ref{['prop:anti_commutator_control']}, assuming $0.99$ visibility of a Werner state for the local states $\sigma_{AA_F}$ and $\tau_{BB_G}$ with a CHSH self-test. For the shared state $\rho_{AB}$, we relate the visibility to the quantum bit-error rate and enforce the resulting BB84 statistics as constraints on a Werner state with visibility $0.96$. A single switch corresponds to applying the relaxed marginal constraint on Alice’s side only, while two switches apply it to both parties.

Theorems & Definitions (7)

• theorem 1
• Proposition 1: Anti-commutator control
• theorem 1
• proof
• proof : Proof of Proposition \ref{['prop:anti_commutator_control']}
• theorem 2
• proof