Number-Theoretic Nature of Communication in Quantum Spin Systems

TL;DR

This paper characterizes the specific lengths of unmodulated quantum spin chains that enable nearly perfect quantum state transfer, revealing a deep connection between quantum communication and number theory.

Contribution

It provides an exact characterization of chain lengths allowing high-fidelity transfer, linking quantum dynamics to prime numbers and powers of two.

Findings

Perfect transfer occurs if and only if chain length n=p-1, 2p-1 with p prime, or n=2^{m}-1.

The results connect quantum communication capabilities to the arithmetic structure of integers.

The study offers a number-theoretic perspective on quantum spin system dynamics.

Abstract

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.