  Topological phases exhibit edge responses that are robust against
deformations and defects. Such phases have recently been realized in stochastic
systems, which model a large class of biological and chemical phenomena.
However, general theoretical principles are lacking for these systems, such as
the relation between the bulk topological invariant and observed edge
responses, i.e. the celebrated bulk-edge correspondence. We show that contrary
to established topological phases, stochastic systems require non-reciprocal
(or non-Hermitian) transitions to have edge responses, which holds across all
dimensions and geometries. For both equilibrium and non-equilibrium steady
states, we demonstrate that stochastic edge responses grow dramatically with
non-reciprocity while the quantum version plateaus. Having established the
necessity and importance of non-reciprocity for stochastic edge states, we
point out fundamental theoretical and physical consequences of this result.
Lastly, we present a novel mechanism by which non-reciprocity engenders robust
edge currents in stochastic systems. Our work establishes the crucial role of
non-reciprocal interactions in permitting robust responses in active and living
matter.
