The generation and manipulation of strong entanglement and Einstein-Podolsky-Rosen (EPR) steering in macroscopic systems are outstanding challenges in modern physics. Especially, the observation of asymmetric EPR steering is important for both its fundamental role in interpreting the nature of quantum mechanics and its application as resource for the tasks where the levels of trust at different parties are highly asymmetric. Here, we study the entanglement and EPR steering between two macroscopic magnons in a hybrid ferrimagnet-light system. In the absence of light, the two types of magnons on the two sublattices can be entangled, but no quantum steering occurs when they are damped with the same rates. In the presence of the cavity field, the entanglement can be significantly enhanced, and strong two-way asymmetric quantum steering appears between two magnons with equal dispassion. This is very different from the conventional protocols to produce asymmetric steering by imposing additional unbalanced losses or noises on the two parties at the cost of reducing steerability. The essential physics is well understood by the unbalanced population of acoustic and optical magnons under the cooling effect of cavity photons. Our finding may provide a novel platform to manipulate the quantum steering and the detection of bi-party steering provides a knob to probe the magnetic damping on each sublattice of a magnet.
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