  Theories in physics can provide a kind of map of the physical system under
investigation, showing all of the possible types of behavior which may occur.
Certain points on the map are of greater significance than others, because they
describe how the system responds in a useful or interesting manner. For
example, the point of resonance is of particular importance when timing the
pushes onto a person sat on a swing. More sophisticatedly, so-called
exceptional points have been shown to be significant in optical systems
harbouring both gain and loss, as typically described by non-Hermitian
Hamiltonians. However, expressly quantum points of interest -- be they
exceptional points or otherwise -- arising in quantum photonic systems have
been far less studied. Here we consider a paradigmatic model: a pair of coupled
qubits subjected to an unbalanced ratio of gain and loss. We mark on its map
several flavours of both exceptional and critical points, each of which are
associated with unconventional physical responses. In particular, we uncover
the points responsible for characteristic spectral features and for the sudden
loss of quantum entanglement in the steady state. Our results provide
perspectives for characterizing quantum photonic systems beyond effective
non-Hermitian Hamiltonians, and suggest a hierarchy of intrinsically quantum
points of interest.
