We discuss the analogy between topological entanglement and quantum entanglement, particularly for tripartite quantum systems.We illustrate our approach by first discussing two clearly (topologically) inequivalent systems of three-ring links: The Borromean rings, in which the removal of any one link leaves the remaining two non-linked (or, by analogy, non-entangled); and an inequivalent system (which we call the NUS link) for which the removal of any one link leaves the remaining two linked (or, entangled in our analogy).We introduce unitary representations for the appropriate Braid Group (B3) which produce the related quantum entangled systems. We finally remark that these two quantum systems, which clearly possess inequivalent entanglement properties, are locally unitarily equivalent.
We discuss the analogy between topological entanglement and quantum entanglement, particularly for tripartite quantum systems. We illustrate our approach by first discussing two clearly (topologically) inequivalent systems of three-ring links: The Borromean rings, in which the removal of any one link leaves the remaining two non-linked (or, by analogy, non-entangled); and an inequivalent system (which we call the NUS link) for which the removal of any one link leaves the remaining two linked (or, entangled in our analogy). We introduce unitary representations for the appropriate Braid Group (
B3) which produce the related quantum entangled systems. We finally remark that these two quantum systems, which clearly possess inequivalent entanglement properties, are locally unitarily equivalent.
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
Complete sets of commuting observables (CSCOs) of the form ${\ensuremath{\Sigma}}_{N}={\ensuremath{\Pi}}_{i=1}^{N}{\ensuremath{\sigma}}_{i{\ensuremath{\alpha}}_{i}}\phantom{\rule{0.3em}{0ex}}({\ensuremath{\alpha}}_{i}=x,y,z)$ for an
N-qubit system are extracted by a simple graphic approach. One can construct $2\ifmmode\times\else\texttimes\fi{}{3}^{N}$ sets of operators, each set consisting of
KN commuting ${\ensuremath{\Sigma}}_{N}$, ${K}_{N}={2}^{N\ensuremath{-}1}+1$ for even
N, and ${2}^{N\ensuremath{-}1}$ for odd
N. Any
N functional-independent operators among the
KN operators may be adopted as a CSCO, whose simultaneous eigenstates (SEs) span an orthonormal basis of
N-qubit space. These SEs have reduced density matrix of rank 2 and can be reduced to the Greenberger-Horne-Zeilinger (GHZ) state form of Eq. (2) in suitable representations. The all-versus-nothing demolition of the elements of reality holds for each basis of the form of Eq. (2) for
N-qubit $(N\ensuremath{\geqslant}3)$ systems. ${\ensuremath{\Sigma}}_{N}$ may be considered as the infinitesimal operator of rotational operator $R({\ensuremath{\alpha}}_{1},{\ensuremath{\alpha}}_{2},\dots{}{\ensuremath{\alpha}}_{N})={\ensuremath{\Pi}}_{i=1}^{N}\mathrm{exp}[\ensuremath{-}i\ensuremath{\pi}{\ensuremath{\sigma}}_{i{\ensuremath{\alpha}}_{i}}∕2]$ , whose eigenvalue (signature) $r={e}^{\ensuremath{-}i\ensuremath{\pi}\ensuremath{\alpha}}$, or signature exponent $\ensuremath{\alpha}$, may be equivalently used for characterizing each basis.