  The extension of many-body quantum dynamics to the non-unitary domain has led
to a series of exciting developments, including new out-of-equilibrium
entanglement phases and phase transitions. We show how a duality transformation
between space and time on one hand, and unitarity and non-unitarity on the
other, can be used to realize steady state phases of non-unitary dynamics that
exhibit a rich variety of behavior in their entanglement scaling with subsystem
size -- from logarithmic to extensive to \emph{fractal}. We show how these
outcomes in non-unitary circuits (that are "spacetime-dual" to unitary
circuits) relate to the growth of entanglement in time in the corresponding
unitary circuits, and how they differ, through an exact mapping to a problem of
unitary evolution with boundary decoherence, in which information gets
"radiated away" from one edge of the system. In spacetime-duals of chaotic
unitary circuits, this mapping allows us to uncover a non-thermal volume-law
entangled phase with a logarithmic correction to the entropy distinct from
other known examples. Most notably, we also find novel steady state phases with
\emph{fractal} entanglement scaling, $S(\ell) \sim \ell^{\alpha}$ with tunable
$0 < \alpha < 1$ for subsystems of size $\ell$ in one dimension. These
fractally entangled states add a qualitatively new entry to the families of
many-body quantum states that have been studied as energy eigenstates or
dynamical steady states, whose entropy almost always displays either area-law,
volume-law or logarithmic scaling. We also present an experimental protocol for
preparing these novel steady states with only a very limited amount of
postselection via a type of "teleportation" between spacelike and timelike
slices of quantum circuits.
