Voltage quantum graphs and a Gross-Tucker theorem for quantum graphs

Abstract: A voltage graph is a finite directed graph whose edges are labeled by elements of a finite group $G$. A classical construction of Gross and Tucker associates to every voltage graph with vertex set $V$ a so-called derived graph with vertex set $V \times G$. We generalize their construction to quantum graphs and finite abelian groups. Remarkably, the construction can produce true quantum graphs starting from a classical voltage graph. In this case the obtained quantum graph is quantum isomorphic to a classical graph. As a main result we also prove a quantum version of the Gross-Tucker theorem which characterizes precisely which graphs can be written as derived graphs of voltage graphs.

• The paper introduces voltage quantum graphs and crossed product quantum sets, extending classical voltage-graph constructions to finite noncommutative C*-algebras.
• It proves that derived quantum graphs are quantum-isomorphic across different dual-group actions and establishes a Gross–Tucker characterization under Landstad-type conditions for finite abelian groups.
• It shows that every undirected loopfree quantum graph on $(M_2,2\mathrm{Tr})$ arises from a classical $\mathbb{Z}_2$-voltage graph, while identifying non-abelian extensions and quantum Cuntz–Krieger applications as open problems.

The paper develops a noncommutative generalization of the classical theory of voltage graphs and derived graphs due to Gross and Tucker, working within the framework of finite quantum graphs in the sense of Musto–Reutter–Verdon. The central objects are voltage quantum graphs: $G$-indexed families of quantum adjacency matrices on a quantum set, equivariant under an action of the dual group $\hat{G}$ of a finite abelian group $G$. The associated derived quantum graph lives on a newly introduced crossed product quantum set, and the main results are (i) a quantum isomorphism between derived graphs built from different dual-group actions and (ii) a Gross–Tucker-type characterization of which quantum graphs arise as derived graphs.

Background and motivation

Classically, a voltage graph $(\tilde{\Gamma}, \lambda)$ consists of a finite directed graph together with an edge labeling by elements of a finite group $G$; its derived graph has vertex set $V \times G$, with edges lifted according to the labels. The Gross–Tucker theorem states that a graph is isomorphic to a derived graph if and only if it admits a free group action whose quotient recovers the base graph. Voltage graphs have found applications in covering theory, degree–diameter constructions, automorphism lifting, and the decomposition of graph C*-algebras into crossed products up to stable isomorphism (Kumjian–Pask).

Quantum graphs replace the finite vertex set with a finite-dimensional C*-algebra $B$ equipped with a faithful positive functional $\psi$ satisfying $m m^\ast = \mathrm{Id}_B$ (a "quantum set functional," equivalently a scaled $\delta$-form). A quantum adjacency matrix is a Schur-idempotent, $\hat{G}$0-preserving linear map $\hat{G}$1. The paper adopts this definition, noting its equivalence with the projection-based and quantum-relation-based definitions established elsewhere.

Crossed product quantum sets

A key technical contribution is the notion of a crossed product quantum set. Given a quantum set $\hat{G}$2 and an action $\hat{G}$3 of the dual of a finite abelian group, the functional $\hat{G}$4 on the crossed product C*-algebra is defined by $\hat{G}$5 with $\hat{G}$6. The paper proves that $\hat{G}$7 is a quantum set functional if and only if $\hat{G}$8 is, using explicit formulas for the inner product, comultiplication, and adjoint of the scaled conditional expectation $\hat{G}$9 on the crossed product. For the trivial action, the construction reduces to the tensor product quantum set $G$0.

To invert the construction, the paper combines Landstad duality for finite-dimensional C*-algebras with the above functional analysis: if an action $G$1 admits a unitary representation $G$2 diagonalizing $G$3, and additionally $G$4, then $G$5 as quantum sets, where $G$6. The authors are explicit that Landstad's conditions are strictly stronger than mere freeness of the action; classically they correspond to triviality of the principal bundle, i.e., the existence of a $G$7-equivariant map $G$8.

Voltage and derived quantum graphs

A voltage quantum graph is a family $G$9 of quantum adjacency matrices on $(\tilde{\Gamma}, \lambda)$0 satisfying the equivariance condition $(\tilde{\Gamma}, \lambda)$1. The derived quantum graph is

$(\tilde{\Gamma}, \lambda)$2

where $(\tilde{\Gamma}, \lambda)$3 encodes the group translation on the fiber. The operation $(\tilde{\Gamma}, \lambda)$4 plays the role of intersecting two quantum graphs; the authors caution that in the genuinely noncommutative setting this operation need not preserve quantum adjacency matrices in general — here it does, because of the specific structure of $(\tilde{\Gamma}, \lambda)$5 and the equivariance assumption. The paper verifies that each $(\tilde{\Gamma}, \lambda)$6 and the derived graph are genuine quantum adjacency matrices, and establishes inheritance properties: the derived graph is loopfree if $(\tilde{\Gamma}, \lambda)$7 is loopfree, undirected if $(\tilde{\Gamma}, \lambda)$8, and $(\tilde{\Gamma}, \lambda)$9-regular whenever $G$0.

For pre-simple classical voltage graphs (those whose derived graph is simple), the construction recovers Gross and Tucker's derived graph exactly when the dual action is trivial: the derived quantum graph on $G$1 corresponds precisely to $G$2. This compatibility is demonstrated both abstractly and via an explicit one-vertex, one-loop example over $G$3.

Quantum isomorphism of derived graphs

The first main theorem shows that the choice of dual-group action is invisible at the level of quantum isomorphism:

If $G$4 is a voltage quantum graph with respect to any action $G$5, then $G$6.

The proof constructs an explicit quantum isomorphism $G$7 and checks intertwining of the derived adjacency matrices term by term. Consequently, whenever a classical graph $G$8 arises as a derived graph and the voltage graph carries a non-trivial $G$9-action preserving the color classes of edges, one obtains a genuinely quantum graph quantum-isomorphic to the classical $V \times G$0. This yields a systematic three-step recipe for producing "quantum twins" of classical graphs admitting free abelian group actions.

The quantum Gross–Tucker theorem

The second main result characterizes derived quantum graphs. Let $V \times G$1 be a quantum graph with an action $V \times G$2 satisfying the Landstad conditions plus the extra requirement $V \times G$3. Then the maps

$V \times G$4

form a voltage quantum graph on the fixed-point quantum set $V \times G$5, and $V \times G$6. The proof proceeds through a Fourier-type family $V \times G$7 satisfying a convolution identity inherited from Schur-idempotence of $V \times G$8, from which Schur-idempotence and $V \times G$9-preservation of each $B$0 follow. Notably, well-definedness of $B$1 as an endomap of $B$2 is not automatic and requires a separate argument exploiting equivariance of $B$3 under $B$4.

In the classical case the assumptions collapse exactly to freeness of the action (the $B$5 become trivial), so the theorem specializes to the original Gross–Tucker theorem for finite abelian groups. The restriction to abelian groups is a genuine limitation of the present framework: the authors state that extensions to non-abelian groups would require voltage quantum graphs over quantum groups, which they defer to future work.

All quantum graphs on $B$6 as derived graphs

As an application, the paper shows that every undirected loopfree quantum graph on the quantum set $B$7 arises as the derived quantum graph of a classical $B$8-voltage graph on two vertices, using the swap action of $B$9 on $\psi$0, whose crossed product is $\psi$1. Concretely, the three nonempty graphs classified independently by Matsuda and Gromada — with adjacency matrices $\psi$2, $\psi$3, and $\psi$4 expressed via Pauli matrices — are recovered from three explicit voltage graphs involving loops and multiple edges. By the quantum isomorphism theorem, all of these are quantum isomorphic to classical graphs on four vertices, recovering known results as instances of the general phenomenon. A complementary remark computes the full family of derived graphs obtainable from pre-simple $\psi$5-voltage graphs invariant under the vertex swap, parametrized by three bits $\psi$6.

Limitations and open questions

Several restrictions are acknowledged explicitly. The theory is confined to finite abelian groups, since the construction relies on Pontryagin duality and Landstad's theorem for dual actions; the non-abelian case would require quantum-group-valued voltages. The Landstad freeness conditions are stronger than classical freeness, so the quantum Gross–Tucker theorem applies to a narrower class of actions than its classical counterpart. The definition of undirectedness for non-tracial functionals is ambiguous, and the authors adopt the tracial definition while noting alternatives. Finally, the motivating question of whether Kumjian–Pask's stable isomorphism $\psi$7 extends to quantum Cuntz–Krieger algebras of derived quantum graphs remains open.

Conclusion

The paper extends the Gross–Tucker derived-graph machinery to finite quantum graphs by introducing crossed product quantum sets and voltage quantum graphs, proving that derived graphs depend on the dual action only up to quantum isomorphism, and establishing a quantum Gross–Tucker theorem under Landstad-type hypotheses. The complete realization of the $\psi$8 classification through classical two-vertex voltage graphs demonstrates that the construction produces genuinely noncommutative graphs from purely classical input, with quantum isomorphism to classical graphs as a general structural feature.