  We show that two spatially separated semiconductor quantum dots under
resonant and continuous-wave excitation can be strongly entangled in the
steady-state, thanks to their radiative coupling by mutual interaction through
the normal modes of a photonic crystal dimer. We employ a quantum master
equation formalism to quantify the steady-state entanglement by calculating the
system {\it negativity}. Calculations are specified to consider realistic
semiconductor nanostructure parameters for the photonic crystal dimer-quantum
dots coupled system, determined by a guided mode expansion solution of Maxwell
equations. Negativity values of the order of 0.1 ($20\%$ of the maximum value)
are shown for interdot distances that are larger than the resonant wavelength
of the system. It is shown that the amount of entanglement is almost
independent of the interdot distance, as long as the normal mode splitting of
the photonic dimer is larger than their linewidths, which becomes the only
requirement to achieve a local and individual qubit addressing. Considering
inhomogeneously broadened quantum dots, we find that the steady-state
entanglement is preserved as long as the detuning between the two quantum dot
resonances is small when compared to their decay rates. The steady-state
entanglement is shown to be robust against the effects of pure dephasing of the
quantum dot transitions. We finally study the entanglement dynamics for a
configuration in which one of the two quantum dots is initially excited and
find that the transient negativity can be enhanced by more than a factor of two
with respect to the steady-state value. These results are promising for
practical applications of entangled states at short time scales.
