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We consider how a maximally mixed state evolves under $(1+1)D$ Floquetnon-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 symmetrybreaking transition. We show that this symmetry breaking transition coincideswith different kinds of purification transitions. Gaussian non-unitary circuitsare mixed (not purifying) on both sides of the symmetry breaking transition,while interacting but integrable non-unitary circuits are mixed on thesymmetric side and ``weakly purifying" on the symmetry breaking side. In theweakly purifying phase, the initial mixed state purifies on a time scaleproportional to the system size. We obtain numerically the critical exponentsassociated with the divergence of the purification time at the purificationtransition, which depend continuously on the parameters of the model. Uponadding a symmetric perturbation that breaks integrability, the weakly purifyingphase becomes strongly purifying, purifying in a time independent of the systemsize, 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 weaklypurifying phase.

Antiunitary symmetry breaking and a hierarchy of purification transitions in Floquet non-unitary circuits