Quantum-rigid random quantum graphs

TL;DR

This paper investigates quantum automorphism groups of quantum graphs, showing that both classical and quantum symmetries are generically trivial under broad conditions, extending classical results to the quantum setting.

Contribution

It demonstrates that quantum automorphism groups of quantum graphs are generically trivial, providing a quantum analogue of classical almost-rigidity results.

Findings

Quantum automorphism groups are generically trivial for a broad class of quantum graphs.

Classical symmetry groups of quantum graphs are also generically trivial.

The universal preserver of the quantum adjacency matrix relates to a different quantum automorphism group.

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 $\overline{\mathcal{S}}\le M_n$.