  Topological quantum matter is typically associated with gapped phases and
edge modes protected by the bulk gap. In contrast, recent work (Phys. Rev. B
104, 075132) proposed intrinsically gapless topological phases that, in one
dimension, carry protected edge modes only when the bulk is a gapless Luttinger
liquid. The edge modes of such a topological Luttinger liquid (TLL) descend
from a nonlocal string order that is forbidden in gapped phases and whose
precise form depends on the symmetry class of the system. In this work, we
propose a powerful and unbiased entanglement-based smoking gun signature of the
TLL. In particular, we show that the entanglement entropy profile of a TLL
lacks Friedel oscillations that are invariably present in other gapless one
dimensional phases such as ordinary Luttinger liquids, and argue that their
absence is closely related to a long-ranged string order which is an intrinsic
property of the TLL. Crucially, such a diagnostic is more robust against
numerical errors and relatively easier to measure in experiments as it relies
on the entanglement entropy rather than entanglement spectrum, unlike the
entanglement-based diagnostics of gapped topological phases in one dimension.
