  We present a theoretical study of a hybrid circuit-QED system composed of two
semiconducting charge-qubits confined in a microwave resonator. The qubits are
defined in terms of the charge states of two spatially separated double quantum
dots (DQDs) which are coupled to the same photon mode in the microwave
resonator. We analyze a transport setup where each DQD is attached to
electronic reservoirs and biased out-of-equilibrium by a large voltage, and
study how electron transport across each DQD is modified by the coupling to the
common resonator. In particular, we show that the inelastic current through
each DQD reflects an indirect qubit-qubit interaction mediated by off-resonant
photons in the microwave resonator. As a result of this interaction, both
charge qubits stay entangled in the steady (dissipative) state. Finite shot
noise cross-correlations between currents across distant DQDs are another
manifestation of this nontrivial steady-state entanglement.
