  We show that a continuous-variable (CV) qudit can be constructed using
quasiorthogonal cat states of a bosonic mode, when the phase encoded in each
cat state is chosen appropriately. With the constructed CV qudit and the
discrete-variable (DV) qudit encoded with Fock states, we propose an approach
to generate the hybrid maximally entangled state of a CV qudit and a DV qudit
by using two microwave cavities coupled to a superconducting flux qutrit. This
proposal relies on the initial preparation of a superposition of Fock states of
one cavity and the initial preparation of a cat state of the other cavity.
After the initial state of each cavity is prepared, this proposal requires only
two basic operations, i.e., the first operation employs the dispersive coupling
of both cavities with the qutrit while the second operation uses the dispersive
coupling of only one cavity with the qutrit. The entangled state production is
deterministic and the operation time decreases as the dimensional size of each
qudit increases. In addition, during the entire operation, the coupler qutrit
remains in the ground state and thus decoherence from the qutrit is
significantly reduced. As an example, we further discuss the experimental
feasibility for generating the hybrid maximally entangled state of a DV qutrit
and a CV qutrit based on circuit QED. This proposal is universal and can be
extended to accomplish the same task, by using two microwave or optical
cavities coupled to a natural or artificial three-level atom.
