Phase-sensitive framed-ribbon representation of single-qubit Pauli measurements in linear cluster states

Abstract

We provide a geometric classification of single-qubit projective measurements on one-dimensional linear cluster states within a topological framework. Establishing an explicit correspondence between local measurements and surgery operations on an associated link model, we represent the cluster state as a linear Hopf chain. Computational-basis (Z) measurements act as topological severance for bulk qubits and boundary pruning for end qubits. Transverse-basis (X) measurements remove the measured qubit and stratify the remaining state into a superposition of two disjoint but classically correlated segments. In contrast, lateral-basis (Y) measurements preserve a single continuous spliced chain while generating complex phase factors absent from unframed link descriptions. Although the unframed linking structure already distinguishes X- and Y-basis measurement outcomes geometrically, it cannot differentiate the two possible outcomes within a fixed basis. To resolve this ambiguity, we introduce a framed ribbon representation in which quantum phases are encoded as geometric twists, with chiral plus/minus 90 deg twists representing the phases plus/minus i. The resulting framework provides a phase-sensitive, outcome-resolved geometric description of single-shot Pauli measurements on linear cluster states. The twist angles are geometric labels determined by measurement outcomes and associated by-product operators rather than topological invariants, whereas the underlying linking pattern remains a genuine topological datum. The analysis is restricted to single-shot single-qubit Pauli measurements on one-dimensional cluster states; sequential measurements and classical feedforward are left as open problems.