Quantum gate lower bounds for loss-tolerant position verification

Abstract: Quantum position-verification is a cryptographic task wherein a verifier attempts to establish the location in space of a prover. Recent experiments have implemented a well-studied class of position-verification schemes, known as the $f$-BB84 scenario, but their security under realistic loss and imperfect state preparation remains incompletely understood. We give a new lower bound on any attack on this scheme, in particular proving nearly-linear quantum gate lower bounds on the attacker, even when allowing the attacker to declare a transmission loss of up to $50\%$, allowing for the challenges prepared by the referee to be imperfect, and allowing the quantum messages used in the protocol to be arbitrarily slow. Our results are applicable to recent and upcoming experimental implementations of $f$-BB84, and in particular establish their security under a bounded quantum gate assumption on the attacker. The key ingredient is a tight analytic tradeoff for a lossy BB84 monogamy-of-entanglement game, valid without requiring the attackers' outputs to agree, which replaces observations previously made numerically.