We present protocols for the generation of high-dimensional entangled states of anharmonic oscillators by means of coherent manipulation of light-matter systems in the ultrastrong coupling regime. Our protocols consider a pair of ultrastrong coupled qubit-cavity systems, each coupled to an ancilla qubit, and combine classical pulses plus the selection rules imposed by the parity symmetry. We study the robustness of the entangling protocols under dissipative effects. This proposal may have applications within state-of-art circuit quantum electrodynamics.
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