Perfect quantum state transfer on the Johnson scheme

TL;DR

This paper characterizes all Johnson scheme graphs with perfect quantum state transfer, showing that only Kneser graphs $K(2k,k)$ have this property, and explores conditions for unions of such graphs to admit perfect transfer.

Contribution

It provides a complete characterization of Johnson scheme graphs with perfect state transfer, identifying the Kneser graphs $K(2k,k)$ as unique in this regard.

Findings

Kneser graphs $K(2k,k)$ are the only Johnson scheme graphs with perfect state transfer.

Certain unions of Johnson scheme graphs can also admit perfect state transfer under specific conditions.

The paper advances understanding of quantum state transfer properties in algebraically structured graphs.

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.