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 an $XY$ Hamiltonian) that permit transfer with a 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.

Figure 1

Logarithm of the earliest points in which the fidelity is strictly greater than 0.99 for $2\le n\le 7$. The numbers have been obtained by plotting $|U(t{)}_{1,n}|$ ($n=2,\dots ,7$) and then by analyzing sections of the curves. Clearly 0.99 is an arbitrary choice. Notice the jumps between the pairs (2, 3), (4, 5), (6, 7). Because PGST depends on the positive eigenvalues of $P_n$, this phenomenon may be explained by the fact that $\lfloor n/2\rfloor =\lfloor (n+1)/2\rfloor$ for $n$ even.