Measurability of Wilson loop operators

We show that the nondemolition measurement of a spacelike Wilson loop operator $W(C)$ is impossible in a relativistic non-Abelian gauge theory. In particular, if two spacelike-separated magnetic flux tubes both link with the loop C, then a nondemolition measurement of $W(C)$ would cause electric charge to be transferred from one flux tube to the other, a violation of relativistic causality. A destructive measurement of $W(C)$ is possible in a non-Abelian gauge theory with suitable matter content. In an Abelian gauge theory, many cooperating parties distributed along the loop C can perform a nondemolition measurement of the Wilson loop operator if they are equipped with a shared entangled ancilla that has been prepared in advance. We also note that Abelian electric charge (but not non-Abelian charge) can be transported superluminally, without any accompanying transmission of information.