Towards practical and error-robust quantum position verification

Loss of inputs can be detrimental to the security of quantum position verification (QPV) protocols, as it may allow attackers to not answer on all played rounds, but only on those they perform well on. In this work, we study \textit{loss-tolerant} QPV protocols. We propose a new fully loss-tolerant protocol QPV$_{\textsf{SWAP}}$, based on the SWAP test, with several desirable properties. The task of the protocol, which could be implemented using only a single beam splitter and two detectors, is to estimate the overlap between two input states. By formulating possible attacks as a semi-definite program (SDP), we prove full loss tolerance against unentangled attackers restricted to local operations and classical communication, and show that the attack probability decays exponentially under parallel repetition of rounds. We show that the protocol remains secure even if unentangled attackers are allowed to quantum communicate, making our protocol the first fully loss-tolerant protocol with this property. A detailed analysis under experimental conditions is conducted, showing that QPV$_{\textsf{SWAP}}$ remains fairly robust against equipment errors. We identify a necessary condition for security with errors and simulate one instance of our protocol with currently realistic experimental parameters, gathering that an attack success probability of $\leq10^{-6}$ can be achieved by collecting just a few hundred conclusive protocol rounds.