  Quantum chromodynamics is the theory of the strong interaction between quarks
and gluons; the coupling strength of the interaction, $\alpha_S$, is the least
precisely-known of all interactions in nature. An extraction of the strong
coupling from the radiation pattern within jets would provide a complementary
approach to conventional extractions from jet production rates and hadronic
event shapes, and would be a key achievement of jet substructure at the Large
Hadron Collider (LHC). Presently, the relative fraction of quark and gluon jets
in a sample is the limiting factor in such extractions, as this fraction is
degenerate with the value of $\alpha_S$ for the most well-understood
observables. To overcome this limitation, we apply recently proposed techniques
to statistically demix multiple mixtures of jets and obtain purified quark and
gluon distributions based on an operational definition. We illustrate that
studying quark and gluon jet substructure separately can significantly improve
the sensitivity of such extractions of the strong coupling. We also discuss how
using machine learning techniques or infrared- and collinear-unsafe information
can improve the demixing performance without the loss of theoretical control.
While theoretical research is required to connect the extract topics with the
quark and gluon objects in cross section calculations, our study illustrates
the potential of demixing to reduce the dominant uncertainty for the $\alpha_S$
extraction from jet substructure at the LHC.
