Number-Theoretic Nature of Communication in Quantum Spin Systems

The last decade has witnessed substantial interest in protocols for transferring information on networks of quantum mechanical objects. A variety of control methods and network topologies have been proposed, on the basis that transfer with perfect fidelity --- i.e. deterministic and without information loss --- is impossible through unmodulated spin chains with more than a few particles. Solving the original problem formulated by Bose [Phys. Rev. Lett. 91, 207901 (2003)], we determine the exact number of qubits in unmodulated chains (with XY Hamiltonian) that permit the transfer with fidelity arbitrarily close to 1, a phenomenon called pretty good state transfer. We prove that this happens if and only if the number of nodes is n=p-1, 2p-1, where p is a prime, or n=2^{m}-1. The result highlights the potential of quantum spin system dynamics for reinterpreting questions about the arithmetic structure of integers, and, in this case, primality.