Absence of logarithmic and algebraic scaling entanglement phases due to skin effect

Measurement-induced phase transition in the presence of competition between projective measurement and random unitary evolution has attracted increasing attention due to the rich phenomenology of entanglement structures. However, in open quantum systems with free fermions, a generalized measurement with conditional feedback can induce skin effect and render the system short-range entangled without any entanglement transition, meaning the system always remains in the ``area law'' entanglement phase. In this work, we demonstrate that the power-law long-range hopping does not alter the absence of entanglement transition brought on by the measurement-induced skin effect for systems with open boundary conditions. In addition, for the finite-size systems, we discover an algebraic scaling $S(L, L/4)\sim L^{3/2-p}$ when the power-law exponent $p$ of long-range hopping is relatively small. For systems with periodic boundary conditions, we find that the measurement-induced skin effect disappears and observe entanglement phase transitions among ``algebraic law'', ``logarithmic law'', and ``area law'' phases.