Zero transfer on mixed graphs

Quantum state transfer describes how quantum information moves between vertices in a graph-based network. While perfect state transfer is well-studied, zero transfer—a phenomenon where the transition amplitude between two vertices is identically zero for all time—is less understood. This paper investigates the conditions under which zero transfer occurs in mixed graphs (graphs containing both directed and undirected edges) and specifically analyzes oriented circulant graphs.

The authors utilize the Hermitian adjacency matrix, which allows for the representation of directed edges while maintaining the mathematical properties necessary for quantum walk analysis. By examining the spectral decomposition of this matrix, the researchers establish a general criterion for zero transfer based on the vanishing of spectral idempotents. They then apply these theoretical tools to oriented circulant graphs, using algebraic number theory and computational methods to classify instances of zero transfer for various graph orders.

The study establishes that for any mixed graph, zero transfer between two vertices occurs if and only if the weighted sum of all walks between them is zero for all lengths, or equivalently, if the spectral idempotents vanish for those vertices. For oriented circulant graphs, the authors prove that zero transfer is impossible when the graph order is a prime number, as the eigenvalue 0 always has a multiplicity of one. They also provide a computational classification for small orders, demonstrating that zero transfer can emerge in composite-order graphs, such as those of order 21, where specific symmetry in the connection sets forces the transition amplitude to zero.

Understanding zero transfer is critical for designing robust quantum architectures. By identifying conditions that prevent information leakage or unwanted interactions between specific nodes, this research provides a theoretical framework for "quantum shielding" or path isolation in quantum networks. The results offer a systematic way to predict and avoid (or intentionally create) regions of zero transfer in oriented network topologies.