Routed Bell tests with arbitrarily many local parties

Abstract

Device-independent quantum key distribution (DIQKD) promises cryptographic security based solely on observed quantum correlations, yet its implementation over long distances remains limited. 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 develop a general $C^*$-algebraic for routed DIQKD with multiple switches and arbitrarily many local test parties, with a conservative, state-dependent definition of Eve. Within this framework, we design and analyse four-party routed protocols that locally self-test both Alice as well as Bob, and numerically bound key rates from the full observed statistics. In the parameter regimes considered, adding a fourth party strictly improves certified key rates and lowers the non-zero key threshold. Randomized key-basis switching further amplifies this advantage. Finally, we investigate a self-test-assisted E91-type protocol that continuously interpolates between the device-dependent Shor-Preskill rate and the device-independent rate derived by [Pironio et al., New J. Phys. 11, 045021 (2009)].