Asymmetric graphs with quantum symmetry

TL;DR

The paper addresses how classical and quantum symmetries diverge for finite graphs by constructing an infinite family of graphs with trivial classical automorphism groups but nontrivial quantum automorphism groups, grounded in the duals of homogeneous solution groups from binary linear systems. The core method combines a vertex-colored graph construction $G(M,b)$ whose quantum automorphism group realizes $ ext{Dual}(\Gamma(M,0))$, with a polynomial-time decoloring procedure that preserves quantum isomorphism and $ ext{Qut}$ while removing colors. A key technical advance is proving that the decoloring process, color refinement, and certain graph-modifications preserve the quantum isomorphism $C^*$-algebras, enabling uncolored realizations and robust control of quantum symmetry. The results yield a weak quantum analog of Frucht’s theorem (every finite group can appear as a quantum automorphism group of some graph) and show that for every finite group Γ there exist graphs with Aut$(G)\cong ext{Aut}(G')\cong Γ$ but with different quantum automorphism groups, thereby establishing that quantum symmetry is not determined by classical symmetry and ruling out quantum-excluding groups. Overall, this work provides a framework to generate asymmetric graphs with quantum symmetry, connect solution groups to graph quantum symmetries, and demonstrate the ubiquity and non-determinism of quantum automorphism groups in finite graphs.

Abstract

We present an infinite sequence of finite graphs with trivial automorphism group and non-trivial quantum automorphism group. These are the first known examples of graphs with this property. Moreover, to the best of our knowledge, these are the first examples of any asymmetric classical space that has nontrivial quantum symmetries. Our construction is based on solution groups to (binary) linear systems, as defined by Cleve, Liu and Slofstra in the context of non-local games. We first show that the dual quantum group of every solution group occurs as the quantum automorphism group of some graph, and then construct an infinite sequence of systems whose solution groups are nontrivial perfect groups. This leads to the desired sequence of graphs. In addition to our main result, we prove a number of related results that allow us to answer several open problems from the literature. We prove a weak quantum analog of Frucht's theorem, namely that every finite classical group $Γ$ occurs as the quantum automorphism group of a finite graph. Combined with our main result, this shows that, for every finite group $Γ$, there are graphs $G_1$ and $G_2$ that both have classical automorphism group isomorphic to $Γ$ but one of them has quantum symmetry and the other does not. Therefore, the quantum automorphism group of a graph is never determined by its classical automorphism group, and there do not exist any "quantum excluding groups".

Figures (3)

• Figure 1: A worked example illustrating the steps of the decoloring procedure of \ref{['def:decolor']}. The labels next to the vertices refer to the color of that vertex.
• Figure 2: \ref{['proof:perfecta']} (left) and \ref{['proof:perfectb']} (right), up to swapping $x$ and $y$.
• Figure 3: Left: the vertex-and-edge coloring of the neighborhood $N(y)$ from \ref{['figure1']}. Right: the vertex-and-edge coloring of the neighborhood $N(x)$ from \ref{['figure1']}. Black vertices and edges have unknown color.

Theorems & Definitions (98)

• definition 2.1
• example 2.2: CMQG1
• definition 2.3
• definition 2.4: CMQG1
• definition 2.5: WanSn
• definition 2.6: wang1995free
• definition 2.7: RSsolution
• lemma 2.8
• definition 2.9
• definition 3.1