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 $A(X) = w_0A_0 + \cdots + w_m A_m$, 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 $r=1,2,\cdots,m$, \[w_r = \frac{\pi}{\tau} \sum_{j=r}^m \frac{c_j}{\binom{2j}{j}} \binom{k-r}{j-r}.\] 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 $\pi/2$.

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Johnson 方案加权图上的完美​​状态转移

我们在 Johnson 方案 $\mathcal{J}(n,k)$ 的实加权图上刻画完美状态转移。给定 $\mathcal{J}(n,k)=\{A_1, A_2, \cdots, A_k\}$ 和 $A(X) = w_0A_0 + \cdots + w_m A_m$，我们使用经典数论结果显示，当且仅当 $n=2k$，$m\ge 2^{\lfloor{\log_2(k)} \rfloor}$，并且存在整数$c_1, c_2, \cdots, c_m$ 使得 (i) $c_j$ 是奇数当且仅当 $j$ 是 $2$ 的幂，并且 (ii) 对于 $r=1,2,\cdots,m $, \[w_r = \frac{\pi}{\tau} \sum_{j=r}^m \frac{c_j}{\binom{2j}{j}} \binom{kr}{jr}.\ ] 然后，我们在 $\mathcal{J}(n,k)$ 的未加权图上表征完美状态转移。特别是，我们获得了一个简单的构造，它生成了 $\mathcal{J}(n,k)$ 的所有图，并且在 $\pi/2$ 时刻具有完美的状态转移。