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.
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