Number-Theoretic Nature of Communication in Quantum Spin Systems

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 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.

Quantum spin chains with fractional revival

“…In fact, it would suffice in that perspective to consider situations where FR can happen with probability as close to 1 as desired. This question has been analyzed in the case of full revival and has been referred to as almost perfect state transfer (APST) [26] or pretty good state transfer [27]. One may assume that the isospectral deformations of a chain with APST will lead to chains with almost perfect fractional revival (APFR).…”

Quantum state transfer through a spin chain in a multiexcitation subspace

“…Here PGST means that for every ǫ > 0, there exists time t such that the maximum fidelity is greater than 1 − ǫ. For example, when N = 6, Tmax≈ 298, and Fmax> 0.99 (ǫ = 0.01) [21]. But for a long chain, Tmax will be very long for ǫ = 0.01.…”

“…One of these channels is chosen to be the ground state |0 of a FM chain while the other channel corresponds to the Néel state. In a single excitation subspace, for an unmodulated XY Hamiltonian with length N , it has pretty good state transfer (PGST) between the two ends of the chain if and only if N = p − 1 or N = 2p − 1, where p is prime, or if N = 2 m − 1 [21]. Here PGST means that for every ǫ > 0, there exists time t such that the maximum fidelity is greater than 1 − ǫ.…”

Walk entropies in graphs

“…The study of a graph invariant in one field may also be a result of relevant importance in other areas of physics. This is because graphs are nowadays ubiquitous in many areas of physics such as in problems associated with the Ising, Potts and Hubbard models, in the solution of Feynman integrals in perturbative field theory, in quantum information theory such as quantum error correcting codes (graph states) or arrangements of interacting quantum mechanical particles (spin networks) [3][4][5][6][7], and in many other fields (see [8] and references therein).…”