Perfect cheating is impossible for single-qubit position verification

In quantum position verification, a prover certifies her location by performing a quantum computation and returning the results (at the speed of light) to a set of trusted verifiers. One of the very first protocols for quantum position verification was proposed in (Kent, Munro, Spiller 2011): the prover receives a qubit $Q$ from one direction, receives an orthogonal basis $\{v,v^{\perp }\}$ from the opposite direction, then measures $Q$ in $\{v,v^{\perp }\}$ and broadcasts the result. A number of variants of this protocol have been proposed and analyzed, but the question of whether the original protocol itself is secure has never been fully resolved. In this work we show that there is no perfect finite-dimensional cheating strategy for the original KMS measurement protocol. Our approach makes use of tools from real algebraic geometry.