  The essence of the famed long-range entanglement as revealed in topologically
ordered state is the paradoxical coexistence of short-range correlation and
nonlocal information that cannot be removed through constant-depth local
quantum circuits. Its realization in different quantum states is a focus
research topic in both quantum computation and quantum matter. However, the
proved realizations are subject to the paradigm of topological order (including
its extensions), i.e. via a quantum code structure with macroscopic code
distance. Here, we broaden the knowledge of long-range entangled states by
rigorously proving the coexistence in a new concrete state. The state describes
qudits on the newly experimentally discovered fractal lattice geometry (1.58D)
on which the quantum code structure has been shown not to exist, i.e., there is
no topological order. Our result might reveal a new paradigm for the
realization of the long-range entanglement in many-body quantum states, and
might stimulate new studies connecting quantum information and quantum matter.
