A group-theoretic approach to elimination measurements of qubit sequences
Mark Hillery, Erika Andersson, Ittoop Vergheese
TL;DR
The paper develops a group-theoretic framework for designing elimination measurements, formalized as covariant POVMs where |ψ_g⟩ = Γ(g)|ψ_e⟩ and Π_g = Γ(g)|X⟩⟨X|Γ(g)^{-1} annihilates the corresponding state. It demonstrates constructions for eliminating a single state in two-qubit and multi-qubit settings using abelian groups (e.g., Z2 × Z2) and a non-abelian example with D3, including explicit |X⟩ vectors and the resulting measurement elements. The approach extends to eliminating multiple states by exploiting subgroup cosets, yielding multi-set elimination in three and four-qubit scenarios, with entropic bounds constraining feasibility. An entropic bound based on the Holevo limit provides a quantitative constraint on when such elimination measurements can exist, illustrating a fundamental trade-off between state structure, symmetry, and information extracted. The work broadens the toolkit for anti-distinguishability and anti-discrimination in quantum information tasks, with implications for quantum foundations and cryptography.
Abstract
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. 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. 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.