  A comparative study of entropy dynamics as an indicator of physical behavior
in an open two-state system with balanced gain and loss is presented. We
distinguish the perspective taken in utilizing the conventional framework of
Hermitian-adjoint states from an approach that is based on biorthogonal-adjoint
states and a third case based on an isospectral mapping. In this it is
demonstrated that their differences are rooted in the treatment of the
environmental coupling mode. For unbroken $\mathcal{PT}$ symmetry of the
system, a notable characteristic feature of the perspective taken is the
presence or absence of purity oscillations, with an associated entropy revival.
The description of the system is then continued from its
$\mathcal{PT}$-symmetric pseudo-Hermitian phase into the regime of
spontaneously broken symmetry, in the latter two approaches through a
non-analytic operator-based continuation, yielding a Lindblad master equation
based on the $\mathcal{PT}$ charge operator $\mathcal{C}$. This phase
transition indicates a general connection between the pseudo-Hermitian
closed-system and the Lindbladian open-system formalism through a spontaneous
breakdown of the underlying physical reflection symmetry.
