Fractional revival on oriented Cayley and semi-Cayley graphs over abelian groups

Abstract: Fractional revival (FR), a generalization of perfect state transfer (PST), is a significant phenomenon in quantum state transfer that allows quantum information to be transmitted between two qubits with a certain probability. The existence of FR has been extensively studied on many classes of graphs. However, oriented graphs have not yet been investigated. In this paper, we investigate the existence of proper FR on oriented graphs. We first establish necessary and sufficient conditions for oriented graphs to admit proper FR between strongly cospectral vertices. Furthermore, we prove that oriented Cayley graphs over abelian groups do not admit proper FR, and we subsequently characterize the conditions under which oriented semi-Cayley graphs over abelian groups admit proper FR.

• The paper initially studies fractional revival (FR) on *oriented* graphs, establishing necessary and sufficient conditions for proper FR between strongly cospectral vertices and framing new spectral characteristics for FR.
• Author's confirm that no proper fractional revival (FR) exists on oriented Cayley graphs over finite abelian groups, emphasizing the skew-symmetric adjacency.
• The work complete characteristics of proper FR on oriented semi-Cayley graphs over abelian groups, detailing practical cases and providing constructive examples of graphs with explicit transfer probabilities.
• questions
• What are the practical implications of the bidirectional nature of FR in oriented graphs?
• How does the skew-symmetry of the adjacency matrix $A_\Gamma$ impact the feasibility of FR in oriented graphs?
• In what ways does the character table of a group influence the feasibility of FR in oriented Cayley and semi-Cayley graphs?
• For which types of groups beyond abelian groups, might the authors' results be extendable to study FR?
• Find recent papers about quantum transport phenomena on directed graphs.

Overview and contribution

This paper by Jiang, Liu, and Wang initiates the study of fractional revival (FR) on oriented graphs, a setting previously unexamined for this quantum transport phenomenon. Working with the real skew-symmetric adjacency matrix $A_\Gamma$ as Hamiltonian — so that the transition matrix is $U(t)=\exp(A_\Gamma t)$, which is real unitary — the authors establish three main results: (i) necessary and sufficient spectral conditions for proper FR between strongly cospectral vertices in arbitrary oriented graphs; (ii) a nonexistence theorem showing that oriented Cayley graphs over finite abelian groups admit no proper FR at all; and (iii) a full characterization of proper FR on oriented semi-Cayley graphs $\mathrm{OSC}(G,R,L,S)$ over abelian groups. The paper also extends the theory of strong cospectrality to the oriented setting, where eigenvalues are purely imaginary or zero and eigenvectors are generally complex.

Spectral characterization of FR between strongly cospectral vertices

The paper adapts strong cospectrality to oriented graphs: vertices $u$ and $v$ are strongly cospectral if for each eigenvalue $\lambda_r$ there exists a phase factor $e^{\mathrm{i}\pi q_r(u,v)}$ with $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$, with phases constant across eigenspaces of equal eigenvalue. Strongly cospectral vertices are shown to be cospectral.

The central structural result is that FR in the oriented setting is inherently bidirectional: if $(\alpha,\beta)$-FR occurs from $u$ to $U(t)=\exp(A_\Gamma t)$0 at time $U(t)=\exp(A_\Gamma t)$1 between cospectral vertices, then $U(t)=\exp(A_\Gamma t)$2-FR occurs from $U(t)=\exp(A_\Gamma t)$3 to $U(t)=\exp(A_\Gamma t)$4 at the same time. This follows from unitarity of the real transition matrix together with equality of diagonal entries. The proof forces the entire $U(t)=\exp(A_\Gamma t)$5-column of $U(t)=\exp(A_\Gamma t)$6 to be supported only on $U(t)=\exp(A_\Gamma t)$7 — a rigidity that later drives both nonexistence results.

The main characterization states that $U(t)=\exp(A_\Gamma t)$8-FR with $U(t)=\exp(A_\Gamma t)$9 occurs between strongly cospectral $\mathrm{OSC}(G,R,L,S)$0 at time $\mathrm{OSC}(G,R,L,S)$1 iff for every eigenvalue $\mathrm{OSC}(G,R,L,S)$2 in the support of $\mathrm{OSC}(G,R,L,S)$3:

$\mathrm{OSC}(G,R,L,S)$4

An immediate consequence is that all relevant phase factors must equal $\mathrm{OSC}(G,R,L,S)$5, partitioning the support into sets $\mathrm{OSC}(G,R,L,S)$6 and $\mathrm{OSC}(G,R,L,S)$7. Because $\mathrm{OSC}(G,R,L,S)$8 is skew-symmetric, eigenvalues come in conjugate pairs $\mathrm{OSC}(G,R,L,S)$9, and the authors prove that $u$0 iff $u$1. This symmetry collapses the periodicity condition to a single congruence class: fixing $u$2, FR occurs exactly when $u$3 for all $u$4.

A notable feature of the oriented case is that FR can occur between more than one pair simultaneously — unlike undirected graphs, where Kay's uniqueness theorem restricts PST to a single partner per vertex. The paper does not pursue multi-pair FR systematically here, leaving it as an open question.

Nonexistence of proper FR on oriented abelian Cayley graphs

For $u$5 with $u$6 abelian of order $u$7 and $u$8, the character table $u$9 diagonalizes $v$0, and all vertices are cospectral: $v$1 depends only on $v$2. This translation invariance is fatal for FR. Suppose $v$3-FR occurs from $v$4 to $v$5. By translation invariance, FR also occurs from $v$6 to $v$7, while the bidirectionality theorem gives $v$8. If $v$9, then column $\lambda_r$0 has squared entries $\lambda_r$1, contradicting orthogonality of columns of $\lambda_r$2. If instead $\lambda_r$3, comparing entries yields $\lambda_r$4, forcing $\lambda_r$5.

This negative result is unconditional: no choice of connection set $\lambda_r$6 in any finite abelian group produces proper FR. The obstruction is purely structural — vertex-transitivity combined with skew-symmetric adjacency makes the two-sided FR pattern incompatible with unitarity. It contrasts sharply with the undirected case, where Wang et al. exhibited genuine families of abelian Cayley graphs admitting FR.

Characterization for oriented semi-Cayley graphs

The semi-Cayley construction $\lambda_r$7 has vertex set $\lambda_r$8 with intra-side arcs governed by $\lambda_r$9 and $e^{\mathrm{i}\pi q_r(u,v)}$0 and cross-side arcs from side 0 to side 1 governed by $e^{\mathrm{i}\pi q_r(u,v)}$1. Over an abelian $e^{\mathrm{i}\pi q_r(u,v)}$2, the character table block-diagonalizes the adjacency matrix into $e^{\mathrm{i}\pi q_r(u,v)}$3 blocks

$e^{\mathrm{i}\pi q_r(u,v)}$4

yielding explicit eigenvalues $e^{\mathrm{i}\pi q_r(u,v)}$5 and Kronecker-product eigenprojectors.

Three results structure the analysis:

• No FR within a side: proper FR from $e^{\mathrm{i}\pi q_r(u,v)}$6 to $e^{\mathrm{i}\pi q_r(u,v)}$7 is impossible, via the same three-column unitarity contradiction as in the Cayley case.
• Block form of the transition matrix: if FR occurs from $e^{\mathrm{i}\pi q_r(u,v)}$8 to $e^{\mathrm{i}\pi q_r(u,v)}$9, then necessarily

$$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$0

for a permutation matrix $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$1 determined by $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$2 and an orthogonal matrix $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$3.

• Characterization: writing $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$4, FR from $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$5 to $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$6 requires $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$7 plus explicit exponential constraints on $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$8 involving $$E_{\lambda_r}\mathbf e_u = e^{\mathrm{i}\pi q_r(u,v)} E_{\lambda_r}\mathbf e_v$$9 and $(\alpha,\beta)$0. In terms of strong cospectrality, the conditions become: (a) no character annihilates $(\alpha,\beta)$1; (b) $(\alpha,\beta)$2 for all $(\alpha,\beta)$3; (c) a phase-coherence condition tying equal eigenvalues to consistent values of $(\alpha,\beta)$4. Given strong cospectrality, FR reduces to a partition $(\alpha,\beta)$5 according to the sign of $(\alpha,\beta)$6, with the periodicity conditions $(\alpha,\beta)$7 and $(\alpha,\beta)$8, and $(\alpha,\beta)$9, $u$0 read off from $u$1. Note that condition (b), $u$2 for all $u$3, is derived rather than assumed, and it constrains the admissible pairs $u$4 considerably.

Two concrete constructions confirm feasibility. For $u$5 with $u$6, the graph admits $u$7-FR from $u$8 to $u$9 at time $U(t)=\exp(A_\Gamma t)$00; here the transfer probability $U(t)=\exp(A_\Gamma t)$01 approaches 1 as $U(t)=\exp(A_\Gamma t)$02 grows, though never reaching it for finite $U(t)=\exp(A_\Gamma t)$03. A second example with nonempty $U(t)=\exp(A_\Gamma t)$04, $U(t)=\exp(A_\Gamma t)$05 over $U(t)=\exp(A_\Gamma t)$06 achieves $U(t)=\exp(A_\Gamma t)$07-FR at time $U(t)=\exp(A_\Gamma t)$08, i.e., transfer probability $U(t)=\exp(A_\Gamma t)$09.

These examples show that the index-two-like doubling of the vertex set is what enables FR: the cross-side coupling through $U(t)=\exp(A_\Gamma t)$10 breaks the translation-invariance obstruction that kills the Cayley case. The characterization is complete but conditional — it presumes FR endpoints lie on opposite sides and are strongly cospectral; FR between non-cospectral vertices is explicitly left open.

Limitations and open questions

The paper's results are confined to abelian underlying groups, where character-table diagonalization is available; the nonabelian case is not addressed. The general characterization (Theorem FRNS) applies only to strongly cospectral endpoint pairs, and the authors do not determine whether proper FR can occur without strong cospectrality in oriented graphs. Within the semi-Cayley analysis, the necessity direction of the FR characterization relies on the Fourier inversion argument applied entrywise, which presupposes the FR endpoints have the specific form $U(t)=\exp(A_\Gamma t)$11; whether asymmetric choices of $U(t)=\exp(A_\Gamma t)$12 beyond those satisfying $U(t)=\exp(A_\Gamma t)$13 could still support FR is excluded only under strong cospectrality. Three problems are stated explicitly: existence of FR on oriented Cayley graphs over nonabelian groups; characterization of FR between non-cospectral vertices; and construction of oriented graphs admitting FR between every pair of vertices, the fractional analogue of universal perfect state transfer.

Conclusion

The paper supplies the first systematic treatment of fractional revival on oriented graphs. Its principal findings are a phase-based spectral criterion for FR between strongly cospectral vertices, a definitive impossibility result for oriented abelian Cayley graphs, and a workable, checkable characterization for oriented semi-Cayley graphs over abelian groups, supported by explicit cyclic constructions attaining transfer probabilities up to $U(t)=\exp(A_\Gamma t)$14. The separation between the Cayley and semi-Cayley cases isolates vertex-transitivity as the decisive obstruction, and the stated open problems indicate where the theory must next be extended.