A Tight Lower Bound for the BB84-states Quantum-Position-Verification Protocol

We use the entanglement sampling techniques developed by Dupuis, Fawzi and Wehner to find a lower bound on the entanglement needed by a coalition of cheater attacking the quantum position verification protocol using the four BB84 states in the scenario where the cheaters have no access to a quantum channel but share a (possibly mixed) entangled state $\tilde{\mathrm{\Phi }}$. For a protocol using n qubits, a necessary condition for cheating is that the max- relative entropy of entanglement $E_{\max }(\tilde{\mathrm{\Phi }})\ge n-O(\log n)$. This improves previously known best lower bound by a factor $\sim 4$, and it is essentially tight, since this protocol is vulnerable to a teleportation based attack using $n-O(1)$ ebits of entanglement.