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This research establishes a rigorous mathematical classification of entanglement monotones by linking them to the edge-labelled Cayley graphs of finite Coxeter systems, covering both classical and exceptional families. The work provides exact verification code and data for the largest quotients, such as $E_8/D_7$, enabling precise positivity tests over specific rational fields. By formalizing the structural properties of multipartite pure states under local unitaries, this study advances the theoretical foundations required for analyzing complex quantum entanglement structures.
Monotones from multi-invariants: the Coxeter classification
arXiv:2509.06348v2 Announce Type: replace Abstract: Multi-invariants are multiplicative polynomial invariants of multipartite pure states under local unitaries. They yield entanglement monotones when their contraction graphs satisfy edge-convexity. We classify all connected edge-convex multi-invariants. Up to duplication of entire edge-colour families, their contraction graphs are precisely the standard edge-labelled Cayley graphs of finite Coxeter systems. Necessity follows by identifying edge-reflecting graphs with Coxeter Cayley graphs. For sufficiency, we prove vertex-convexity of parabolic quotients and lift their matrices by induction. This covers all classical and exceptional families. For the largest quotients, $E_8/D_7$ and $H_4/A_3$, wall symmetry reduces the matrices to exact positivity tests over $\mathbb Q$ and $\mathbb Q(\sqrt5)$.
Abstract:Multi-invariants are multiplicative polynomial invariants of multipartite pure states under local unitaries. They yield entanglement monotones when their contraction graphs satisfy edge-convexity. We classify all connected edge-convex multi-invariants. Up to duplication of entire edge-colour families, their contraction graphs are precisely the standard edge-labelled Cayley graphs of finite Coxeter systems. Necessity follows by identifying edge-reflecting graphs with Coxeter Cayley graphs. For sufficiency, we prove vertex-convexity of parabolic quotients and lift their matrices by induction. This covers all classical and exceptional families. For the largest quotients, $E_8/D_7$ and $H_4/A_3$, wall symmetry reduces the matrices to exact positivity tests over $\mathbb Q$ and $\mathbb Q(\sqrt5)$.
Substantially expanded revision. Includes a complete proof of the Coxeter classification of connected edge-convex multi-invariants, including all exceptional cases. Exact verification code and data are provided as ancillary files
arXiv:2509.06348 [quant-ph]
(or arXiv:2509.06348v2 [quant-ph] for this version)
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