  Active agents are capable of exerting nonreciprocal forces upon one another.
For instance, one agent may attract another one, which in turn is repelled.
These antagonistic nonreciprocal interactions have been extensively studied in
classical systems, revealing a wealth of exciting phenomena such as novel phase
transitions and traveling-wave states. Whether these phenomena can originate in
quantum many-body systems is an open issue, and proposals for their realization
are lacking. In this work, we present a model of two species of quantum spins
that interact in an antagonistic nonreciprocal way of the attraction-repulsion
type. We propose an implementation based on two atomic ensembles coupled via
chiral waveguides featuring both braided and non-braided geometries. The spins
are active due to the presence of local gain, which allows them to synchronize.
We show that nonreciprocal interactions deeply affect their synchronization
dynamics. In the thermodynamic limit, this results in a nonreciprocal phase
transition to nonstationary traveling-wave states, associated with spontaneous
breaking of parity-time symmetry. We establish how this symmetry emerges from
the microscopic quantum model. For a finite number of spins, we discuss how
traveling-wave states and nonreciprocal phase transitions are revealed by spin
correlations measurable via the output field of the waveguides. Our work opens
a new avenue to explore nonreciprocal interactions in active quantum matter.
