Directed Graph States in Quantum Systems

• Directed graph states are quantum state representations defined by oriented graphs and non-commutative operations.
• They incorporate various construction methods, including recursive state preparation, density matrix encoding, and gate-sequence protocols.
• These methods advance quantum computation by precisely controlling entanglement and extending standard graph state theory.

Directed graph states are quantum states or state representations whose defining data are supplied by a directed graph, but recent literature uses the term for several non-equivalent constructions. Some works extend the usual graph-state formalism by orienting edges and using non-commutative two-qudit gates; others assign multipartite states to digraphs through commuting controlled operations or through $RXX$ interactions on weighted arcs; still others treat a weighted digraph as a source of a density matrix via Laplacian-type matrices, or as the support of KMS equilibrium states on a graph $C^*$-algebra. A distinct line of work uses a directed graph only as an organizing device for families of Kirkwood–Dirac-classical states, rather than as an encoding of a single state (Jesus et al., 27 Jan 2026, Simone et al., 15 May 2025, Gnatenko, 2024, Adhikari et al., 2012, Moutuou et al., 2024, Cai et al., 14 Mar 2026).

1. Terminological scope and relation to standard graph states

In the standard qubit formalism, a graph state is defined on a simple undirected graph by

$$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$

Because $CZ^{(a,b)}=CZ^{(b,a)}$ and all $CZ$ gates commute, the state depends only on the undirected graph. A recent extraction framework from two-dimensional cluster states remains entirely within this undirected stabilizer setting and explicitly does not introduce directed graph states (Jesus et al., 27 Jan 2026, Freund et al., 13 May 2025).

The main current meanings of the term are therefore best distinguished by the role played by direction.

Framework	State associated to the graph	Role of direction
$GR$-based directed graph states (Jesus et al., 27 Jan 2026)	Recursive multipartite qudit state $\ket{\Gamma_n}$	Edge orientation and vertex ordering determine non-commuting gate order
Ordering-free digraph states (Simone et al., 15 May 2025)	$$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$	Orientation fixes control and target
Directed weighted-network states (Gnatenko, 2024)	$$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$	Arc weights and in/out structure govern local entanglement
Laplacian digraph states (Adhikari et al., 2012)	$\rho_G=K(G)/\operatorname{Tr}(K(G))$	Complex weights and loops enter $C^*$0
Graph-algebra KMS states (Moutuou et al., 2024)	KMS state on a Toeplitz–Cuntz–Krieger algebra	Directed paths govern equilibrium information flow
KD directed-graph organization (Cai et al., 14 Mar 2026)	Path-selected family of KD-classical states	Graph organizes convex geometry, not a single graph state

A common source of confusion is the assumption that any directed graph state must be a stabilizer state obtained by orienting the edges of an ordinary graph state. The literature summarized above shows otherwise: some constructions are genuinely gate-based state-preparation rules, some are density-matrix encodings of digraphs, and some use a directed graph only to structure a state space.

2. Orientation-dependent graph-state formalism and exchange symmetry

A direct extension of the graph-state formalism appears in the construction based on a non-commutative two-qudit gate $C^*$1. The starting point is the standard $C^*$2-based theory, where a graph state is fully symmetric under particle permutations if and only if the underlying graph is the complete graph $C^*$3. The same analysis also proves that fully antisymmetric states cannot be obtained from any $C^*$4-based graph construction, because basis states such as $C^*$5 retain nonzero amplitude and are invariant under permutations (Jesus et al., 27 Jan 2026).

To go beyond that limitation, the oriented-edge gate is defined by

$C^*$6

with subtraction modulo $C^*$7. In contrast to $C^*$8, one has $C^*$9, and gates on different edges generally do not commute. Orientation is therefore intrinsic rather than decorative: a directed edge specifies which qudit occupies which slot of $$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$0, and vertex ordering fixes the gate sequence.

The recursive antisymmetric construction is

$$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$1

with local dimension $$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$2. When the graph is the complete directed graph with a suitable hierarchical orientation, the resulting directed graph state $$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$3 coincides with $$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$4. For odd $$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$5, this gives the totally antisymmetric $$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$6-qudit state

$$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$7

so that

$$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$8

For even $$\ket{G}=\prod_{(a,b)\in E} CZ^{(a,b)}\ket{+}^{\otimes N}.$$9, the same construction is antisymmetric only under permutations of the first $CZ^{(a,b)}=CZ^{(b,a)}$0 subsystems (Jesus et al., 27 Jan 2026).

This formalism establishes a precise duality: complete undirected graphs with commuting symmetric $CZ^{(a,b)}=CZ^{(b,a)}$1 gates generate fully symmetric states, whereas complete directed graphs with appropriate orientations and the non-commutative $CZ^{(a,b)}=CZ^{(b,a)}$2 gate generate fully antisymmetric states for odd numbers of qudits. The directed graph is therefore not merely a labeled version of an undirected graph; it is the combinatorial record of ordered interactions.

3. Ordering-free directed graph states and degree-determined entanglement

A different notion of directed graph state is defined on a digraph $CZ^{(a,b)}=CZ^{(b,a)}$3 by

$CZ^{(a,b)}=CZ^{(b,a)}$4

where

$CZ^{(a,b)}=CZ^{(b,a)}$5

and

$CZ^{(a,b)}=CZ^{(b,a)}$6

The construction is restricted to the ordering-free case, namely

$CZ^{(a,b)}=CZ^{(b,a)}$7

so that no edge-ordering ambiguity survives in the product (Simone et al., 15 May 2025).

Entanglement is measured by the Entanglement Distance derived from the Fubini–Study metric,

$CZ^{(a,b)}=CZ^{(b,a)}$8

Writing $CZ^{(a,b)}=CZ^{(b,a)}$9 and $CZ$0 for out-degree and in-degree, and $CZ$1 for total degree, the main theorem is

$CZ$2

The local contribution satisfies

$CZ$3

Two consequences are central. First, the entanglement distance is invariant under vertex relabeling, because it depends only on the degree multiset. Second, it is insensitive to the distinction between incoming and outgoing edges: the quantum state $CZ$4 does depend on orientation, since control and target are different, but this particular entanglement measure depends only on total degree. For a $CZ$5-regular digraph,

$CZ$6

and for a complete graph on $CZ$7 vertices,

$CZ$8

A frequent misconception is therefore ruled out in this setting: orientation does not automatically imply orientation-sensitive entanglement. What is orientation-sensitive is the state construction; what is orientation-blind here is the chosen Fubini–Study-based measure (Simone et al., 15 May 2025).

4. Directed weighted-network states and local geometric entanglement

A third construction maps an arbitrary directed weighted graph $CZ$9 to a multi-qubit state by

$GR$0

with

$GR$1

and separable input

$GR$2

Vertices correspond to qubits, and ordered weighted arcs correspond to two-qubit $GR$3 interactions with parameters $GR$4 (Gnatenko, 2024).

For a vertex $GR$5, the paper separates the incident structure into incoming vertices $GR$6, outgoing vertices $GR$7, and bidirected neighbors $GR$8. The geometric measure of entanglement of qubit $GR$9 with the rest is

$\ket{\Gamma_n}$0

and, in the symmetric-weight homogeneous-input case,

$\ket{\Gamma_n}$1

This makes the graph dependence explicit: indegree, outdegree, and the corresponding arc weights jointly determine the local entanglement.

In the unweighted undirected reduction, the dependence collapses to the ordinary degree $\ket{\Gamma_n}$2: $\ket{\Gamma_n}$3 and for $\ket{\Gamma_n}$4, $\ket{\Gamma_n}$5,

$\ket{\Gamma_n}$6

The paper also gives an operational protocol for estimating $\ket{\Gamma_n}$7. One measures $\ket{\Gamma_n}$8 directly in the computational basis, $\ket{\Gamma_n}$9 after applying $$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$0, and $$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$1 after applying $$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$2. The three-qubit chain with arcs $$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$3 and $$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$4 yields, for $$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$5,

$$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$6

and the analytic formulas were validated on AerSimulator with $$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$7 shots per data point (Gnatenko, 2024).

5. Laplacian and operator-algebraic constructions

In the Laplacian framework, a weighted directed graph $$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$8 is mapped directly to a density matrix rather than to a gate sequence. The basic prescription is

$$\ket{G}=\prod_{(a,b)\in L}U_{ab}\ket{\phi}^{\otimes M}$$9

where $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$0 is the combinatorial Laplacian, $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$1 the signless Laplacian, and $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$2 signed Laplacians adapted to negative or mixed loop weights. Edge weights may be complex, loops may be present, and purity is equivalent to $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$3. For loopless weighted digraphs, purity occurs iff $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$4 or $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$5; for digraphs with nonnegative loops, purity occurs iff the graph consists of one looped vertex and the remaining isolated vertices. Bell, GHZ, W, cluster, Chi, Brown, and $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$6-qubit Chi states are represented within this formalism, and all entangled pure states in the paper share the same combinatorial motif: all nonzero edge weights lie in a complete subgraph, and each vertex of that subgraph carries a loop of weight $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$7, where $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$8 is the number of vertices in the complete subgraph (Adhikari et al., 2012).

A different graph-to-state map arises from graph $$\ket{\psi_G}=\prod_{(i,j)\in A}RXX_{ij}(\phi_{ij})\ket{\psi_0}$$9-algebras. For a finite directed multigraph with source and range maps $\rho_G=K(G)/\operatorname{Tr}(K(G))$0, the Toeplitz–Cuntz–Krieger algebra is generated by vertex projections $\rho_G=K(G)/\operatorname{Tr}(K(G))$1 and edge partial isometries $\rho_G=K(G)/\operatorname{Tr}(K(G))$2, with gauge dynamics

$\rho_G=K(G)/\operatorname{Tr}(K(G))$3

KMS states of this algebra are then interpreted as equilibrium states of information flow on the directed network. If $\rho_G=K(G)/\operatorname{Tr}(K(G))$4 is the adjacency matrix and $\rho_G=K(G)/\operatorname{Tr}(K(G))$5 its spectral radius, KMS states exist for $\rho_G=K(G)/\operatorname{Tr}(K(G))$6, there is a unique KMS state at $\rho_G=K(G)/\operatorname{Tr}(K(G))$7, and there are no KMS states for $\rho_G=K(G)/\operatorname{Tr}(K(G))$8. In the $\rho_G=K(G)/\operatorname{Tr}(K(G))$9 synaptic network studied in detail, the critical inverse temperature is $C^*$00. Pure KMS states supported on a single neuron determine outgoing interaction profiles across all directed paths, with path contributions weighted by $C^*$01 (Moutuou et al., 2024).

These two constructions are often grouped together only because both assign quantum states to directed graphs. Formally they are very different: one is spectral-graph-theoretic and finite-dimensional, the other operator-algebraic and thermodynamic.

6. KD directed-graph geometry and other adjacent usages

In the Kirkwood–Dirac setting, the graph does not encode a state through entangling gates, adjacency matrices, or Laplacians. Instead, it organizes the geometry of KD-classical states for two bases related by a discrete Fourier transform. For every factorization $C^*$02, one has a family of KD-classical pure states $C^*$03, and the directed graph $C^*$04 has one vertex $C^*$05 for each divisor $C^*$06 of $C^*$07, with each vertex identified with the corresponding family of pure projectors. Edges connect factorizations that differ by multiplication or division by a prime factor of $C^*$08, and the orientation is chosen relative to a coprime factorization $C^*$09. For any path from $C^*$10 to $C^*$11,

$C^*$12

The crucial point is explicit in the source: the graph does not encode a particular quantum state via adjacency operators or stabilizers; it is a combinatorial organizing tool for families of KD-classical pure states and for the convex slices generated by directed paths (Cai et al., 14 Mar 2026).

The phrase also appears outside quantum-state theory proper. In machine learning, DirEgo2Token and DirGraphSSM define nodewise latent states on directed graphs by sequentializing $C^*$13-hop predecessor ego-graphs and applying state space model kernels through message passing; the reported results are state-of-the-art on three representative directed graph learning tasks, competitive on two additional tasks, and $C^*$14 to $C^*$15 faster in training than existing state-of-the-art models (She et al., 17 Sep 2025). In spectral graph theory, the “state” of a directed graph can denote its Laplacian/metric configuration, with effective-resistance-preserving symmetrization yielding

$C^*$16

and preserving the square-root effective-resistance metric (Fitch, 2018).

This suggests that “directed graph states” is fundamentally contextual. In one context it denotes oriented-edge entangling constructions; in another it denotes density matrices or equilibrium states attached to a digraph; in another it denotes a graph-structured decomposition of a state space. Any technical use therefore has to specify which of these constructions is intended, because the shared phrase masks substantial differences in ontology, algebraic structure, and operational meaning.