  In superconducting circuit architectures for quantum computing, microwave
resonators are often used both to isolate qubits from the electromagnetic
environment and to facilitate qubit state readout. We analyze the full counting
statistics of photons emitted from such driven readout resonators both in and
beyond the dispersive approximation. We calculate the overlap between
emitted-photon distributions for the two qubit states and explore strategies
for its minimization with the purpose of increasing fidelity of
intensity-sensitive readout techniques. In the dispersive approximation and at
negligible qubit relaxation, both distributions are Poissonian, and the overlap
between them can be easily made arbitrarily small. Nondispersive terms of the
Hamiltonian generate squeezing and the Purcell decay with the latter effect
giving the dominant contribution to the overlap between two distributions.
