  There are no clean samples in nature. Therefore, when we come to discuss the
entanglement properties of quantum materials, the effects of disorder must be
taken into account. This question is of particular interest for high-entangled
states, such as quantum spin liquids, which lie outside the Landau paradigm for
classifying phases of matter. In this work, we explore what
experimentally-accessible measures, in the form of concurrence, residual tangle
and quantum Fisher information, can teach us about the entanglement in the
presence of disorder. As a representative example, we consider the
Tomonaga-Luttinger liquids (TLL) and disorder-driven random singlet (RS) states
found in antiferromagnetic quantum spin chains. Using quantum Fisher
information and residual tangle, we demonstrate that both TLL and RS states
exhibit multi-partite entanglement. In the case of the RS state, we attribute
this to entanglement localized below a crossover length scale. We further show
that the order of disorder average matters in calculating measures like
concurrence, and that this can lead to false inferences when interpreting
experiment. None the less, correctly interpreted, these witnesses provide
useful information about the effects of disorder. We explore how information
about the central charge of the TLL can be extracted from the low-temperature
behavior of concurrence, and conjecture that this analysis can be extended to
the effective central charge of the RS state. Finally, we establish how RS and
TLL states can be distinguished through the growth of multi-partite
entanglement, as witnessed by the equal-time structure factor. These results
establish that, used carefully, experiments based on entanglement witnesses can
provide important information about quantum spin systems in the presence of
disorder.
