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arXiv:2608.09076

Knot your average qutrit: Measurement-induced entanglement splitting and the cabling dictionary for GHZ and W States

Sougata Bhattacharyya, Sovik Roy

quant-phSubmitted Aug 10, 2026• Announced Aug 11, 2026• 0 citations
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Abstract

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

Comments
27 pages (manuscript), 17 pages (Appendix), 5 figures

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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 WWW 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
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In [Science 340:1205, (2013)], via entanglement polytopes Michael Walter et al. obtained a finite yet systematic classification of multi-particle entanglement. It is well known that under SLOCC, pure states of three (four) qubits are partitioned into six (nine) families. In this paper,we present a LU invariant and an entanglement measures for the GHZ SLOCC class of three qubits, and partition states of the GHZ SLOCC class of three qubits into ten families and each family into two subfamilies under LU. We give a necessary and sufficient condition for the uniqueness of the generalized Schmidt decomposition for the GHZ SLOCC class.
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We propose an alternative classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for a given number of rings. To determine all different possibilities, we develop a formalism that associates any link to a polynomial, with each polynomial thereby defining a distinct equivalence class. To demonstrate the use of this classification scheme, we choose qubit quantum states as our example of physical system. A possible procedure to obtain qubit states from the polynomials is also introduced, providing an example state for each link class. We apply the formalism for the quantum systems of three and four qubits and demonstrate the potential of these tools in a context of qubit networks.
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