Geometric model of quantum navigation during (anti-)search on a plane

A model of joint random walk of two agents on an infinite plane is considered. The agents possess no means of mutual classical communication, but have access to quantum entanglement resource which is used according to a pre-arranged protocol. Depending on the details of the protocol, an effective force of attraction or repulsion emerges between the two agents. The emergence of this force from quantum entanglement is interpreted in terms of spherical or hyperbolic geometries for attraction or repulsion, respectively.