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.
We investigate how network geometry and excitation sector separately control pairwise entanglement decay in asymmetric multipartite $W$ states. To disentangle these effects, we introduce an analytically tractable $N$-qubit generalization of the asymmetric Lohmayer geometry and its complementary-excitation partner, yielding inequivalent vertex-base (VB) and base-base (BB) pair classes that can be compared directly with symmetric $W$-state references. We derive closed-form concurrence dynamics under representative one-sided noise models and find that, within either excitation sector, the VB concurrence has exactly the same noise dependence as the corresponding symmetric reference, preserving a noise-independent proportional advantage wherever both remain entangled. The amplitude-damping reordering previously identified for the three-qubit Lohmayer state is therefore a cross-sector effect rather than an intrinsic fragility of the VB geometry. In contrast, the BB pair exhibits a genuine same-sector structural fragility, with lower entanglement-sudden-death thresholds than the VB pair under depolarizing noise and, in the $(N-1)$-excitation sector, under amplitude damping. The results establish network geometry, excitation sector, and noise symmetry as distinct ingredients governing pairwise entanglement robustness in asymmetric quantum networks.
S. Bhattacharyya, Sovik Roy, Fatih Ozaydin· 0 citations
Multipartite entanglement is conventionally classified by state families viz. family of GHZ and W class of states, with each family expected to behave differently under measurement. We show that, at least for the question of how entanglement splits after a single-particle measurement, this is not the division that matters for qutrits. Extending the Aravind's correspondence (which models entanglement as topological linking, and projective measurement as physically cutting a ring from an interlinked configuration\cite{aravind1997}) from qubits to qutrits, we derive the complete measurement-induced entanglement splitting of the GHZ type qutrit state i.e |GHZ_3>and of the full family of symmetric W class qutrit states, six two same - one different states i.e |{W_{p,p,q}^{sym}}>and one all - different state i.e. |W_{0,1,2}>, under both the computational basis (CB) and the mutually unbiased bases (MUBs), obtaining exact eigenvalues and Schmidt ranks for every outcome in every case. We see that the |W_{0,1,2}>state behaves similarly as |GHZ_3>state, a single, outcome-independent residual rank in each basis, while the |{W_{p,p,q}^{sym}>states alone show probability-weighted, outcome-dependent behaviour. The relevant structural line is therefore repeated-index versus all-different-index bag structure, not GHZ class versus $W$ class. We express this classification using a \textit{two-strand cabling} extension of Aravind's \textit{ring-and-link} picture. This is needed because the qutrit residual Schmidt rank (R) takes three values, R belonging to {1,2,3}, rather than the qubit binary (i.e. R belonging to {1,2}). We are explicit throughout that this cabling dictionary is a labeling convention built to reproduce an independently computed Schmidt rank, not a topological invariant derived from the link diagrams themselves, and we discuss what would be needed to close that gap
S. Bhattacharyya, Sovik Roy· 0 citations
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