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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 vectorspace of initial bipartite states whose time development is either constant oronly dependent on $t + t'$ and is invariant to which quantum walker subsystemgoes with each graph. The set of null spaces corresponding with a set of$W_{G_i,G_j}$ have interesting relations as subspaces, intersections betweensubspaces, and subspaces of intersections. These relations are depicted asEuler diagrams for labeled graphs of three and four vertices.

Indistinguishable quantum walks on graphs relative to a bipartite quantum walker