  We consider how a maximally mixed state evolves under $(1+1)D$ Floquet
non-unitary circuits with an antiunitary symmetry that squares to identity,
that serves as a generalized $\mathcal{PT}$ symmetry. Upon tuning a parameter,
the effective Hamiltonian of the Floquet operator demonstrates a symmetry
breaking transition. We show that this symmetry breaking transition coincides
with different kinds of purification transitions. Gaussian non-unitary circuits
are mixed (not purifying) on both sides of the symmetry breaking transition,
while interacting but integrable non-unitary circuits are mixed on the
symmetric side and ``weakly purifying" on the symmetry breaking side. In the
weakly purifying phase, the initial mixed state purifies on a time scale
proportional to the system size. We obtain numerically the critical exponents
associated with the divergence of the purification time at the purification
transition, which depend continuously on the parameters of the model. Upon
adding a symmetric perturbation that breaks integrability, the weakly purifying
phase becomes strongly purifying, purifying in a time independent of the system
size, for sufficiently large system size. Our models have an extra $U(1)$
symmetry that divides the Hilbert space into different magnetization sectors,
some of which demonstrate logarithmic scaling of entanglement in the weakly
purifying phase.
