Quantum-rigid random quantum graphs
Alexandru Chirvasitu, Piotr M. Sołtan, Mateusz Wasilewski
TL;DR
This work compares two quantum automorphism notions for quantum graphs on finite matrix algebras and shows that both are generically trivial when the operator system ${\mathcal{S}}\le M_n$ ranges over a nonempty Zariski-open set with $2\le d\le n^2-3$. The authors deploy a geometric, algebraic approach via the Grassmannian and Zariski topology to prove generic rigidity, reducing to diagonally acting quantum groups and employing an inductive strategy across dimensions and matrix size; an auxiliary result ties the universal preservers of the quantum adjacency matrix to the conjugate operator system ${\overline{\mathcal{S}}}$. Base-case analyses for small $n$ (notably $n=3$ and $n=4$) seed the induction, with computer-assisted verification supporting the generic claim in several concrete instances (e.g., a $7\times 7$ example). The findings extend classical rigidity results to a fully quantum setting and illuminate the structural relationship between different formulations of quantum symmetries in finite-dimensional quantum graphs, with implications for understanding symmetry in quantum information contexts.
Abstract
A quantum graph $\mathcal{G}$ housed by a matrix algebra $M_n$ can be encoded as an operator system $\mathcal S=\mathcal{S}_{\mathcal{G}}\le M_n$. There are two sensible notions of quantum automorphism group for any such: $\mathrm{Qut}(\mathcal G)$, capturing the quantum symmetries of the adjacency matrix $A:M_n\to M_n$ attached to $\mathcal{G}$, and $\mathrm{Qut}(\mathcal S\le M_n)$, the quantum group acting universally on $M_n$ so as to preserve its $C^*$ structure, standard trace, and subspace $\mathcal{S}\le M_n$. The two quantum groups coincide classically, but diverge in general. We nevertheless show that both are generically trivial in the sense that they are so for $\mathcal{S}\le M_n$ ranging over a non-empty Zariski-open set under all reasonable dimensional constraints on $\dim \mathcal{S}$ and $n$. This extends analogous prior results by the first and third authors to the effect that classical symmetry groups of still-quantum graphs are generically trivial, and offers a fully quantum counterpart to the familiar probabilistic almost-rigidity of finite graphs. An auxiliary result sheds some light on the relationship between the two notions of quantum automorphism group, identifying the universal preserver of the quantum adjacency matrix of $\mathcal{G}$ with the quantum automorphism group not of $\mathcal{S}\le M_n$, but rather of the complex conjugate $\overline{\mathcal{S}}\le M_n$.