A group-theoretic approach to elimination measurements of qubit sequences

Most measurements are designed to tell you which of several alternatives have occurred, but it is also possible to make measurements that eliminate possibilities and tell you an alternative that did not occur. Measurements of this type have proven useful in quantum foundations and in quantum cryptography. Here we show how group theory can be used to design such measurements. This requires that the set of states being considered possesses a symmetry described by a group. After some general considerations, we focus on the case of measurements on two-qubit states that eliminate one state. We then move on to construct measurements that eliminate two three-qubit states and four four-qubit states. The sets of states eliminated constitute cosets of a subgroup. A condition that constrains the construction of elimination measurements is then presented. Finally, in an appendix, we briefly consider the case of elimination measurements with failure probabilities and an elimination measurement on n-qubit states.