Perfect state transfer on weighted graphs of the Johnson scheme

Abstract

We characterize quantum perfect state transfer on real-weighted graphs of the Johnson scheme ${\mathcal {J}}(n,k)$, which represent spin networks with non-nearest neighbor couplings. Given ${\mathcal {J}}(n,k)=\{A_{1},A_{2},\ldots ,A_{k}\}$ and $A(X) = w_0A_0 + \cdots + w_m A_m$, we show 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 $c_{1},c_{2},\ldots ,c_{m}$ such that

• (i) $c_j$ is odd if and only if j is a power of 2, and

• (ii) for $r=1,2,\ldots ,m$,

  $$\begin{aligned} w_r = \frac{\pi }{\tau } \sum _{j=r}^m \frac{c_j}{\left( {\begin{array}{c}2j\\ j\end{array}}\right) } \left( {\begin{array}{c}k-r\\ j-r\end{array}}\right) . \end{aligned}$$

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 $\pi /2$.