  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 $\tilde{\Phi}$. For a
protocol using n qubits, a necessary condition for cheating is that the max-
relative entropy of entanglement $E_{\max}(\tilde{\Phi})\ge n-O(\log n)$. This
improves previously known best lower bound by a factor $\sim4$, and it is
essentially tight, since this protocol is vulnerable to a teleportation based
attack using $n-O(1)$ ebits of entanglement.
