Indistinguishable quantum walks on graphs relative to a bipartite quantum walker

Abstract: A distinguishability operator is defined for the continuous-time quantum walk (CTQW) of a bipartite quantum walker on two simply connected graphs, $W_{G_i,G_j} = U_{G_i}\left(t\right) \otimes U_{G_j}\left(t'\right) - U_{G_j}\left(t'\right) \otimes U_{G_i}\left(t\right)$, where $U_{G_i}\left(t\right)$ is the unitary CTQW operator for a labeled graph $G_i$ over a time interval $t$. The null space of $W_{G_i,G_j}$ defines the vector space of initial bipartite states whose time development is either constant or only dependent on $t + t'$ and is invariant to which quantum walker subsystem goes with each graph. The set of null spaces corresponding with a set of $W_{G_i,G_j}$ have interesting relations as subspaces, intersections between subspaces, and subspaces of intersections. These relations are depicted as Euler diagrams for labeled graphs of three and four vertices.