Local $ε$-uniform mixing in continuous quantum walks

Abstract

Let $X$ be a weighted graph and $M$ be its adjacency, Laplacian or signless Laplacian matrix. In a continuous quantum walk on $X$, local $ε$-uniform mixing occurs at vertex $u$ if the $u$th column of the matrix $U(t)=e^{itM}$ can be made arbitrarily close to a vector whose all entries have equal magnitude. Using the spectral and combinatorial properties of $X$, we derive necessary conditions for local $ε$-uniform mixing to occur in $X$. This includes an inequality involving all entries of each eigenvector of $M$, as well as an upper bound on the degree of vertex $u$ when $M$ is the Laplacian or signless Laplacian matrix. We use these necessary conditions to rule out local $ε$-uniform mixing in numerous classes of graphs, most of which are non-regular. We also show that almost all planar graphs (resp., trees) contain a vertex that does not admit local $ε$-uniform mixing for any assignment of edge weights. Furthermore, we prove if $X$ has $n$ vertices and admits local $ε$-uniform mixing at a vertex contained in a subgraph with a twin, then the number of vertices of this twin subgraph must be at least $\sqrt{n}$. In particular, we establish that a graph on $n\geq 5$ vertices does not admit local $ε$-uniform mixing at a vertex with a twin.

Figures (3)

• Figure 1: Graphs $X$ (left) and $Y$ (right) with false twin subgraphs $G$ and $H$ isomorphic to $P_3$
• Figure 2: Graphs with true twin subgraphs $G$ and $H$ isomorphic to $K_2$; $G$ and $H$ remain true twin subgraphs after inserting the dashed edges
• Figure 3: A graph $X$ with pendent vertices $u$ and $w$ sharing a common neighbor $v$.

Theorems & Definitions (98)

• Proposition 1
• proof
• Corollary 1
• Proposition 2
• proof
• Corollary 2
• Corollary 3
• Proposition 3
• proof
• Proposition 4