Spreading entanglement through pairwise exchange interactions

Abstract

Quantum state synthesis typically aims to convert an initial direct-product state into a highly entangled target state. As a benchmark problem, we consider the task of spreading one excitation among N two-level atoms or qubits. Starting from an initial state where one qubit is excited, we seek a target state where all qubits have the same excitation amplitude—a generalized-W state. This target is to be reached by suitably chosen pairwise exchange interactions. For example, we may have a setup where any pair of qubits can be brought into proximity for a controllable period of time. We describe three protocols that accomplish this task, each with $N-1$ tightly constrained steps. In the first, one atom acts as a flying qubit that sequentially interacts with all others. In the second, qubits interact pairwise in sequential order. In these two cases, the required interaction times follow a pattern with an elegant geometric interpretation. They correspond to angles within the spiral of Theodorus—a construction known for more than two millennia. The third protocol follows a divide-and-conquer approach—dividing equally between two qubits at each step. For large N, the flying-qubit protocol yields a total interaction time that scales as $\sqrt{N}$, while the sequential approach scales linearly with N. For the divide-and-conquer approach, the time has a lower bound that scales as $\log N$—achievable when multiple exchange interactions can take place in parallel. We argue that this is the optimal solution that requires the least time. For any protocol that achieves this task, the phase differences in the final state cannot be independently controlled. As an example, we show that a W state (where all phases are equal) cannot be generated by pairwise exchange.