  For solid-state spin systems, the collective spin motion in a single crystal
embodies multiple magnetostatic modes. Recently, it was found that the
cross-Kerr interaction between the higher-order magnetostatic mode and the
Kittel mode introduces a new operable degree of freedom. In this work, we
propose a scheme to entangle two magnon modes via the cross-Kerr nonlinearity
when the bias field is inhomogeneous and the system is driven. Quantum
entanglement persists at the steady state, as demonstrated by numerical results
using experimentally feasible parameters. Furthermore, we also demonstrate that
entangled states can survive better in the system where self-Kerr and
cross-Kerr nonlinearities coexist. Our work provides insights and guidance for
designing experiments to observe entanglement between different degrees of
freedom within a single ferrimagnetic crystal. Additionally, it may stimulate
potential applications in quantum information processing using spintronic
devices.
