This work sits off to the side of quantum networks: nearby are studies on distributing entanglement between single qubits and on quantum memory in ensembles. But here an ensemble of thousands of atoms acts as a single multidimensional qudit, and entanglement is handed out not one qubit at a time but in macroscopic portions at once. This sets the protocol apart from neighboring work, where entanglement is limited to a single ebit.
The authors model a chain of M ensembles, each containing N qubits. Neighboring ensembles interact through collective spin operators, then the intermediate nodes are measured in the x basis, which creates entanglement between the end nodes. For large N, a Fock-space truncation is used: only states within a window of width c√N around the mean excitation number are kept. This reduces the dimension from (N+1)^M to roughly (c√N+1)^M. Decoherence is modeled as collective S_z dephasing via the Lindblad equation.
In the absence of noise, entanglement grows as log2(N+1)/2, which is comparable to the maximum possible for two symmetric ensembles. The entanglement curve has a fractal 'devil's crevice' structure with dips at rational fractions of π. Under moderate dephasing (γ ≤ 10⁻²), entanglement at 'magic times' (for example, t=π/2, 2π/3) is preserved almost without loss, while strong noise suppresses it monotonically. Numerically, sizes of N=10⁶ without noise and N=30 with noise were reached, which is orders of magnitude larger than the previous N=20 and N=3.
The work shows that macroscopic entanglement is not a fragile exotic: it survives at realistic noise levels and scales with ensemble size. This distinguishes it from neighboring studies, where entanglement is limited to small systems. The protocol does not depend on N, since all operations are symmetric under qubit permutations, which makes it efficient for large ensembles.
One could explore alternative ways to generate entanglement between neighboring nodes, for example, quantum nondemolition measurements, which already yield maximum entanglement for two ensembles. Another direction is applications of distributed macroscopic entanglement in distributed quantum computing and quantum sensing.
The results will influence the development of quantum repeaters and quantum networks, where atomic ensembles serve as nodes with quantum memory. They are also important for quantum metrology and distributed sensors that use collective spin states.
Test the protocol with other types of elementary entanglement generation, for example, via quantum nondemolition measurements. Extend the simulation to a larger number of nodes M and account for other noise sources, such as spontaneous emission.
In quantum information science, the question of scaling quantum networks to a global level remains open. This work links the problem of decoherence of macroscopic states to the practical implementation of quantum repeaters, showing that moderate noise does not destroy entanglement at key moments.
The entanglement curve resembles a 'devil's staircase': dips occur exactly at moments when the interaction time is a rational fraction of π. As the number of atoms grows, these dips become ever denser, like the teeth of a comb with a spacing of 1/N.