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

Abstract

We provide an explicit geometric classification of single-qubit projective measurements on one-dimensional (1D) linear cluster states (LCS) within a topological framework. By establishing an explicit geometrical correspondence between local measurements and topological surgery operations on an associated link model i.e. a measurement-surgery correspondence, we represent the cluster state as a linear Hopf chain. Within this model, measurements in the computational ( Z) basis act as topological severance in case of bulk measurements while boundary pruning happens for end measurements of qubits. In contrast, transverse ( X) basis measurements remove the measured qubit and, rather than fusing its neighbours into a single splice, induce a geometric stratification of the residual state into a superposition of two disjoint, correlated segments joined only through real-valued classical correlations. We show that lateral ( Y) basis measurements instead preserve a single continuous spliced chain while generating intrinsically complex phase factors that are not captured by unframed link models. The unframed linking pattern already distinguishes the X- and Y-basis outcomes from one another by shape; what it cannot resolve is which of the two possible outcomes occurred within a fixed basis, since both members of either outcome pair share an identical unframed diagram. To resolve this residual ambiguity, we introduce a framed ribbon representation in which quantum phases are encoded as geometric twists, with chiral ±90∘ twists corresponding to the phases ±i. This framing yields a phase-sensitive, outcome-resolved geometric description of single-shot Pauli measurements on LCS. We stress that the twist angles introduced here are not topological invariants: they are geometric labels fixed operationally by the measurement outcome and by the resulting by-product operator, whereas the linking pattern of the underlying unframed link is a genuine topological datum. The present work is deliberately restricted to single-shot single-qubit Pauli measurements on 1D LCS; composition rules for sequential measurements and for classical feedforward are not established here and are identified as the principal open problem.