  The photon-blockade breakdown bistability can be intuitively explained
invoking the energy spectrum of the interacting qubit-mode system. Yet, the
neoclassical solution of the driven-dissipative Jaynes-Cummings model has been
shown to capture several key aspects of the phenomenon. In this paper, we set
out to compare a fully quantum solution with the neo- and semi-classical
solutions. Although the neoclassical theory is founded on the assumption of a
pure partial state for the qubit, it is not simply the $\gamma \to 0$ limit of
the semiclassical theory; the semi- vs. neo-classical duality being a case of
non-commutativity of limits. Furthermore, we show that the neoclassical
predictions still hold in case of a small qubit decay. Tracing the bistable
behavior for different detunings, we show that it is robust over a significant
range of $\Delta$ values. We demonstrate that the aptitude of the neoclassical
description is founded on the high quantum purity of the bright state of the
photon-blockade breakdown bistability, which sharply differentiates this
phenomenology from conventional optical bistability. It is thereby demonstrated
that driven-dissipative dynamics can produce closely separable pure steady
states in an interacting bipartite.
