Perfect quantum state transfer on the Johnson scheme

Abstract: For any graph $X$ with the adjacency matrix $A$, the transition matrix of the continuous-time quantum walk at time $t$ is given by the matrix-valued function $\mathcal{H}X(t)=\mathrm{e}^{itA}$. We say that there is perfect state transfer in $X$ from the vertex $u$ to the vertex $v$ at time $\tau$ if $|\mathcal{H}_X(\tau){u,v}| = 1$. It is an important problem to determine whether perfect state transfers can happen on a given family of graphs. In this paper we characterize all the graphs in the Johnson scheme which have this property. Indeed, we show that the Kneser graph $K(2k,k)$ is the only class in the scheme which admits perfect state transfers. We also show that, under some conditions, some of the unions of the graphs in the Johnson scheme admit perfect state transfer.