Abstract
An s-pair state in a graph is a quantum state of the form \(\textbf{e}_u+s\textbf{e}_v\), where u and v are vertices in the graph and s is a nonzero complex number. If \(s=-1\) (resp., \(s=1\)), then such a state is called a pair state (resp. plus state). In this paper, we develop the theory of perfect s-pair state transfer in continuous quantum walks, where the Hamiltonian is taken to be the adjacency, Laplacian or signless Laplacian matrix of the graph. We characterize perfect s-pair state transfer in complete graphs, cycles and antipodal distance-regular graphs admitting vertex perfect state transfer. We construct infinite families of graphs with perfect s-pair state transfer using quotient graphs and graphs that admit fractional revival. We provide necessary and sufficient conditions such that perfect state transfer between vertices in the line graph relative to the adjacency matrix is equivalent to perfect state transfer between the plus states formed by corresponding edges in the graph relative to the signless Laplacian matrix. Finally, we characterize perfect state transfer between vertices in the line graphs of Cartesian products relative to the adjacency matrix.
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Acknowledgements
We thank David Feder, Chris Godsil, Alastair Kay, Christopher van Bommel and Christino Tamon for useful discussions. A. Chan gratefully acknowledges the support of the NSERC Grant No. RGPIN-2021-03609. S. Kim is supported in part by the Fields Institute for Research in Mathematical Sciences and NSERC. S. Kirkland is supported by NSERC grant number RGPIN-2019-05408. H. Monterde is supported by the University of Manitoba Faculty of Science and Faculty of Graduate Studies. S. Plosker is supported by NSERC Discovery grant number RGPIN-2019-05276, the Canada Research Chairs Program grant number 101062, and the Canada Foundation for Innovation grant number 43948.
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Kim, S., Monterde, H., Ahmadi, B. et al. A generalization of quantum pair state transfer. Quantum Inf Process 23, 369 (2024). https://doi.org/10.1007/s11128-024-04574-9
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DOI: https://doi.org/10.1007/s11128-024-04574-9