  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.
