  We propose a scheme for generating and controlling entangled coherent states
(ECS) of magnons, i.e. the quanta of the collective spin excitations in
magnetic systems, or phonons in mechanical resonators. The proposed hybrid
circuit architecture comprises a superconducting transmon qubit coupled to a
pair of magnonic Yttrium Iron Garnet (YIG) spherical resonators or mechanical
beam resonators via flux-mediated interactions. Specifically, the coupling
results from the magnetic/mechanical quantum fluctuations modulating the qubit
inductor, formed by a superconducting quantum interference device (SQUID). We
show that the resulting radiation-pressure interaction of the qubit with each
mode, can be employed to generate maximally-entangled states of magnons or
phonons. In addition, we numerically demonstrate a protocol for the preparation
of magnonic and mechanical Bell states with high fidelity including realistic
dissipation mechanisms. Furthermore, we have devised a scheme for reading out
the prepared states using standard qubit control and resonator field
displacements. Our work demonstrates an alternative platform for quantum
information using ECS in hybrid magnonic and mechanical quantum networks.
