Perfect State Transfer on Weighted Graphs of the Johnson Scheme

Abstract

We characterize perfect state transfer on real-weighted graphs of the Johnson scheme $\mathcal{J}(n,k)$. Given $\mathcal{J}(n,k)=\{A_1,A_2,\cdots ,A_k\}$ and $A(X)=w_0{A}_0+\cdots +w_m{A}_m$, we show, using classical number theory results, that $X$ has perfect state transfer at time $\tau$ if and only if $n=2k$, $m\ge 2^{\lfloor {\log }_2(k)\rfloor }$, and there are integers $c_1,c_2,\cdots ,c_m$ such that (i) $c_j$ is odd if and only if $j$ is a power of $2$, and (ii) for $r=1,2,\cdots ,m$, \[w_r = \fracπτ \sum_{j=r}^m \frac{c_j}{\binom{2j}{j}} \binom{k-r}{j-r}.\] We then characterize perfect state transfer on unweighted graphs of $\mathcal{J}(n,k)$. In particular, we obtain a simple construction that generates all graphs of $\mathcal{J}(n,k)$ with perfect state transfer at time $\pi /2$.

16 pages