Title: Decoherence-free mechanism to protect long-range entanglement against decoherence

Abstract: Quantum spin models with variable-range interactions can exhibit certain quantum characteristics that a short-ranged model cannot possess. By considering the quantum XYZ model whose interaction strength between different sites varies either exponentially or polynomially, we report the creation of long-range entanglement in dynamics both in the absence and presence of system-bath interactions. Specifically, during closed dynamics, we determine a parameter regime from which the system should start its evolution so that the resulting state after quench can produce a high time-averaged entanglement having low fluctuations. Both in the exponential and power-law decays, it occurs when the magnetic field is weak and the interactions in the z-direction are nonvanishing. When part of the system interacts with the bath repeatedly or is attached to a collection of harmonic oscillators along with dephasing noise in the z-direction, we observe that long-range entanglement of the subparts which are not attached with the environment remains constant with time in the beginning of the evolution, known as freezing of entanglement, thereby demonstrating a method to protect long-range entanglement. We find that the frozen entanglement content in any length and the time up to which freezing occurs called the freezing terminal to follow a complementary relation for all ranges of interactions. However, we find that for a fixed range of entanglement, there exists a critical value of interaction length which leads to the maximum freezing terminal.