  Lyman-$\alpha$ (Ly$\alpha$) is a powerful astrophysical probe. Not only is it
ubiquitous at high redshifts, it is also a resonant line, making Ly$\alpha$
photons scatter. This scattering process depends on the physical conditions of
the gas through which Ly$\alpha$ propagates, and these conditions are imprinted
on observables such as the Ly$\alpha$ spectrum and its surface brightness
profile. In this work, we focus on a less-used observable capable of probing
any scattering process: polarization. We implement the density matrix formalism
of polarization into the Monte Carlo radiative transfer code tlac. This allows
us to treat it as a quantum mechanical process where single photons develop and
lose polarization from scatterings in arbitrary gas geometries. We explore
static and expanding ellipsoids, biconical outflows, and clumpy multiphase
media. We find that photons become increasingly polarized as they scatter and
diffuse into the wings of the line profiles, making scattered Ly$\alpha$
polarized in general. The degree and orientation of Ly$\alpha$ polarization
depends on the kinematics and distribution of the scattering HI gas. We find
that it generally probes spatial or velocity space asymmetries and aligns
itself tangentially to the emission source. We show that the mentioned
observables, when studied separately, can leave similar signatures for
different source models. We conclude by revealing how a joint analysis of the
Ly$\alpha$ spectra, surface brightness profiles, and polarization can break
these degeneracies and help us extract unique physical information on galaxies
and their environments from their strongest, most prominent emission line.
