  Recent years have witnessed a surge of research on the non-Hermitian skin
effect (NHSE) in one-dimensional lattices with finite-range couplings. In this
work, we show that the long-range couplings that decay as $1/l^{\alpha}$ at
distance $l$ can fundamentally modify the behavior of NHSE and the scaling of
quantum entanglement in the presence of nonreciprocity. At $\alpha=0$, the
nonlocality of couplings gives rise to the scale-free skin modes, whose
localization length is proportional to the system size. Increasing the exponent
$\alpha$ drives a complex-to-real spectral transition and a crossover from a
scale-free to constant localization length. Furthermore, the scaling of
nonequilibrium steady-state entanglement entropy exhibits a subextensive law
due to the nonlocality and the complex spectrum, in contrast to an area law
arising from NHSE. Our results provide a theoretical understanding on the
interplay between long-range couplings and non-Hermiticity.
