  Local excitations as carriers of quantum information spread out in the system
in ways governed by the underlying interaction and symmetry. Understanding this
phenomenon, also called quantum scrambling, is a prerequisite for employing
interacting systems for quantum information processing. The character and
direction dependence of quantum scrambling can be inferred from the
out-of-time-ordered commutators (OTOCs) containing information on correlation
buildup and entanglement spreading. Employing OTOC, we study and quantify the
directionality of quantum information propagation in oxide-based helical spin
systems hosting a spin-driven ferroelectric order. In these systems,
magnetoelectricity permits the spin dynamics and associated information content
to be controlled by an electric field coupled to the emergent ferroelectric
order. We show that topologically nontrivial quantum phases, such as chiral or
helical spin ordering, allows for electric-field controlled anisotropic
scrambling and a direction-dependent buildup of quantum correlations. Based on
general symmetry considerations, we find that starting from a pure state (e.g.,
the ground state) or a finite temperature state is essential for observing
directional asymmetry in scrambling. In the systematic numerical studies of
OTOC, we quantify the directional asymmetry of the scrambling and verify the
conjectured form of the OTOC around the ballistic wavefront. The obtained
direction-dependent butterfly velocity $v_{\mathrm{B}}(\mathbf{n})$ provides
information on the speed of the ballistic wavefront. In general, our
calculations show an early-time power-law behavior of OTOC, as expected from an
analytic expansion for small times. The long-time behavior of OTOC reveals the
importance of (non-)integrability of the underlying Hamiltonian as well as the
implications of conserved quantities such as the $z$-projection of the total
spin.
