A generalization of quantum pair state transfer

TL;DR

This paper extends the theory of quantum state transfer to s-pair states in graphs, characterizing conditions for perfect transfer across various graph families and matrices, with implications for quantum communication networks.

Contribution

It introduces a comprehensive framework for perfect s-pair state transfer in graphs using different matrices and constructs new graph families with this property.

Findings

Characterization of perfect s-pair state transfer in complete graphs, cycles, and antipodal distance-regular graphs.

Construction of infinite graph families with perfect s-pair state transfer.

Conditions linking perfect state transfer in line graphs to that in original graphs.

Abstract

An $s$-pair state in a graph is a quantum state of the form $\mathbf{e}_u+s\mathbf{e}_v$, where $u$ and $v$ are vertices in the graph and $s$ is a non-zero 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.