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arXiv:2609.12725

Fractional revival in complementary prisms of graphs

Sarojini Mohapatra, Hiranmoy Pal

math.COquant-phSubmitted Sep 11, 2026• Announced Sep 14, 2026
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Abstract

The complementary prism $G\overline{G}$ of a graph $G$ is obtained from the disjoint union of $G$ and its complement $\overline{G}$ by adding an edge between each vertex $a$ in $G$ and its copy $a'$ in $\overline{G}.$ This paper explores a general framework for studying fractional revival with respect to real symmetric matrices with a block structure. The framework is then used to show that, for a fixed state $\mathbf{u}$ in $G$ orthogonal to the all-one vector, the complementary prism $G\overline{G}$ exhibits fractional revival from the state $[\mathbf{u},\mathbf{0}]^T$ with respect to the adjacency, Laplacian, and signless Laplacian matrices. We further characterize perfect pair state transfer in the complementary prism of a complete graph and establish the existence of perfect pair and plus state transfer in the complementary prism of a complete bipartite graph.

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The complementary prism GG‾G\overline{G}GG of a graph GGG is obtained from the disjoint union of GGG and its complement G‾\overline{G}G by adding an edge between each vertex aaa in GGG and its copy a′a'a′ in G‾.\overline{G}.G. This paper explores a general framework for studying fractional revival with respect to real symmetric matrices with a block structure. The framework is then used to show that, for a fixed state u\mathbf{u}u in GGG orthogonal to the all-one vector, the complementary prism GG‾G\overline{G}GG exhibits fractional revival from the state [u,0]T[\mathbf{u},\mathbf{0}]^T[u,0]T with respect to the adjacency, Laplacian, and signless Laplacian matrices. We further characterize perfect pair state transfer in the complementary prism of a complete graph and establish the existence of perfect pair and plus state transfer in the complementary prism of a complete bipartite graph.
We investigate fractional revival in graphs with respect to the adjacency, Laplacian, and signless Laplacian matrices. We observe that, under certain conditions, fractional revival is preserved under graph complementation. Then we establish a connection between fractional revival in a graph and in its double cover, and obtain a complete characterization of pretty good plus state transfer in cycles and their complements. This leads to characterizations of pretty good vertex state transfer in weighted paths with potential.
Given a graph GGG with vertex set V(G)={v1,v2,…,vn1}V(G)=\{v_1,v_2,\ldots,v_{n_1}\}V(G)={v1​,v2​,…,vn1​​} and a graph HHH of order n2n_2n2​, the vertex complemented corona, denoted by G∘~HG\tilde{\circ}{H}G∘~H, is the graph produced by copying HHH n1n_1n1​ times, with the iii-th copy of HHH corresponding to the vertex viv_ivi​, and then adding edges between any vertex in V(G)∖{vi}V(G)\setminus\{v_{i}\}V(G)∖{vi​} and any vertex of the iii-th copy of HHH. The present article deals with quantum state transfer of vertex complemented coronae concerning signless Laplacian matrix. Our research investigates conditions in which signless Laplacian perfect state transfer exists or not on vertex complemented coronae. Additionally, we also provide some mild conditions for the class of graphs under consideration that allow signless Laplacian pretty good state transfer.
Abstract. Pure states correspond to one-dimensional subspaces of [Formula: see text] represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state [Formula: see text] admits PST with another real pure state [Formula: see text], (ii) every connected graph admits PST between real pure states, and (iii) for any pair of real pure states [Formula: see text] and [Formula: see text] and for any time [Formula: see text], there exists a real symmetric matrix [Formula: see text] such that [Formula: see text] and [Formula: see text] admit PST relative to [Formula: see text] at time [Formula: see text]. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all [Formula: see text]-vertex simple unweighted graphs, the least minimum PST time between real pure states is attained by any join graph for the Laplacian case, while it is attained by the join of an empty graph and a complete graph of appropriate sizes for the adjacency case. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at none . [Formula: see text]
We initiate the study of approximate quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of approximate fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give a complete characterizations of when approximate fractional revival can occur in paths and in cycles.
We initiate the study of pretty good quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of pretty good fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give complete characterizations of when pretty good fractional revival can occur in paths and in cycles.
Perfect (s,r)(s,r)(s,r)-state transfer
Much work has been done in the last two decades on the topic of quantum state transfer in a quantum spin network. One can model such a system of interacting qubits using an undirected graph, and studying vertex-to-vertex dynamics. This setup has recently been relaxed to allow for dynamics between linear combinations of two vertex states, i.e.\ from u=ea+seb\mathbf u = \mathbf e_a + s \mathbf e_bu=ea​+seb​ to μ=eα+reβ\mathbf μ=\mathbf e_α + r \mathbf e_βμ=eα​+reβ​, where r=sr=sr=s is either −1-1−1 (which corresponds to pair state transfer) or +1+1+1 (which corresponds to plus state transfer), or more recently r=sr=sr=s is taken to be any real number (which corresponds to sss-pair state transfer). Here, we broaden the investigation of sss-pair state transfer to \textit{perfect (s,r)(s,r)(s,r)-state transfer}, which is perfect state transfer from u=ea+seb\mathbf u = \mathbf e_a + s \mathbf e_bu=ea​+seb​ to μ=eα+reβ\mathbf μ=\mathbf e_α + r \mathbf e_βμ=eα​+reβ​ (up to some dilation) where r,s∈Cr,s\in \mathbb Cr,s∈C. We identify infinite families of graphs with perfect (s,r)(s,r)(s,r)-state transfer and provide characterizations of cases when ∣r∣=∣s∣|r|= |s|∣r∣=∣s∣ and when ∣r∣≠∣s∣|r|\neq |s|∣r∣=∣s∣, showing situations when the degree of entanglement between vertex states is preserved, and when it is not preserved. The latter is particularly important as it represents perfect state transfer from an entangled pair of qubits to another one where the degree of entanglement need not be the same\mdash in fact, it can be set up so as to ``boost'' (increase) entanglement. We provide an algorithm that finds the vector with two nonzero entries that maximizes the fidelity of transfer for a fixed time ttt starting from a given sss-pair state u\mathbf uu. Finally, we provide a sensitivity analysis, with respect to readout time errors, of perfect (s,r)(s,r)(s,r)-state transfer.