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The essence of the famed long-range entanglement as revealed in topologicallyordered state is the paradoxical coexistence of short-range correlation andnonlocal information that cannot be removed through constant-depth localquantum circuits. Its realization in different quantum states is a focusresearch topic in both quantum computation and quantum matter. However, theproved realizations are subject to the paradigm of topological order (includingits extensions), i.e. via a quantum code structure with macroscopic codedistance. Here, we broaden the knowledge of long-range entangled states byrigorously proving the coexistence in a new concrete state. The state describesqudits 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 isno topological order. Our result might reveal a new paradigm for therealization of the long-range entanglement in many-body quantum states, andmight stimulate new studies connecting quantum information and quantum matter.

A new realization of the long-range entanglement: fractality replacing the topological order