Polymer additives are known to cause significant reduction in turbulent friction drag and reduce the energy dissipation rate of fluid transport. This effect is however bounded by a universal upper limit the maximum drag reduction (MDR) asymptote that does not change with polymer properties. Understanding MDR remains an important unsolved problem in the areas of turbulence and non-Newtonian fluid mechanics. Dynamical trajectories on the boundary in state space between laminar and turbulent plane channel flow - edge states - are computed for Newtonian and viscoelastic fluids. Viscoelasticity has a negligible effect on the properties of these solutions and at least at a low Reynolds number their mean velocity profiles correspond closely to experimental observations for polymer solutions in the MDR regime. These results confirm the existence of weak turbulence states that cannot be suppressed by polymer additives explaining the fact that there is an upper limit for polymer-induced drag reduction.