Indistinguishable quantum walks on graphs relative to a bipartite quantum walker

TL;DR

This paper introduces a distinguishability operator for bipartite continuous-time quantum walks on graphs, revealing invariant initial states and their subspace relations through geometric representations.

Contribution

It defines a new operator for bipartite quantum walks and characterizes the invariant state subspaces and their relations using geometric diagrams.

Findings

Null space characterizes invariant initial states.

Subspace relations are depicted as Euler diagrams.

Invariant states depend only on combined time t + t'.

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.