Geometry of Quantum Networks and Quantum Network Routing
A quantum network is a physical arrangement of spins in, say, a nano structure and the spins interact in a way specified by a coupling energy Hamiltonian. The application consists in encoding the message as excitation of one of the spins, the "transmitter," and read out the message with acceptable fidelity, or Information Transfer Capacity (ITC), at another spin, the "receiver." Our contribution has been the geometrization of the ITC; that is, the definition of a quantum mechanical distance that reflects the ITC. It turns out that this distance is completely different from the distance emanating from the physical arrangement of the spins in the nano-device. This observation has been the stepping stone of our research.
A significant difference between classical and quantum networks is that the metric in the latter—the so-called maximum transfer probability—has no classical counterpart, leading to the strange result that spin networks have an anti-core (as opposed to the core of a classical network), which seems to repel all excitation paths (as opposed to attracting all message paths creating congestion in classical networks).
It turns out that the core vs. anti-core behaviour is a corollary of the curvature of the spin network. Classical networks are Gromov hyperbolic with a nontrivial Gromov boundary, creating a gravity center (core), where most of the geodesics pass. Early investigation of spin networks have revealed that they are Gromov hyperbolic, but with a trivial Gromov boundary, resulting in an anti-gravity centre, a point that repels geodesics. This phenomenon is still in need of clarification; in particular, the physical difficulty of communicating with the anti-core center calls for some control, which in turn changes the geometry, leading to the concept of dynamical routing in quantum networks by curvature control.
Finally, it has been assumed thus far that the spin network is "closed," whereas most of the quantum system are "open," that is, they are subject to the decoherence effect of the environment. How does the geometry evolve when the network is subject to decoherence? For how long does the chain remain coherent? How could we fight the decoherence with appropriate control actions?