  Non-Hermitian quantum many-body systems are attracting widespread interest
for their exotic properties, including unconventional quantum criticality and
topology. Here we study how quantum information and correlations spread under a
quantum quench generated by a prototypical non-Hermitian spin chain. Using the
mapping to fermions we solve exactly the problem and compute the entanglement
entropy and the correlation dynamics in the thermodynamic limit. Depending on
the quench parameters, we identify two dynamical phases. One is characterized
by rapidly saturating entanglement and correlations. The other instead presents
a logarithmic growth in time, and correlations spreading faster than the
Lieb-Robinson bound, with collapses and revivals giving rise to a modulated
light-cone structure. Here, in the long-time limit, we compute analytically the
entanglement entropy that we show to scale logarithmically with the size of the
cut, with an effective central charge that we obtain in closed form. Our
results provide an example of an exactly solvable non-Hermitian many-body
problem that shows rich physics including entanglement and spectral
transitions.
