We study Gaussian quantum steering and its asymmetry in the background of a Schwarzschild black hole. We present a Gaussian channel description of quantum state evolution under the influence of the Hawking radiation. We find that thermal noise introduced by Hawking effect will destroy the steerability between an inertial observer Alice and an accelerated observer Bob who hovers outside the event horizon, while it generates steerability between Bob and a hypothetical observer anti-Bob inside the event horizon. Unlike entanglement behaviors in curved spacetime, here the steering from Alice to Bob suffers from a "sudden death" and the steering from anti-Bob to Bob experiences a "sudden birth" with increasing Hawking temperature. We also find that the Gaussian steering is always asymmetric and the maximum steering asymmetry cannot exceed ln 2, which means the state never evolves to an extremal asymmetry state. Furthermore, we obtain the parameter settings that maximize steering asymmetry and find that (i) s = arccosh() is the critical point of steering asymmetry, and (ii) the attainment of maximal steering asymmetry indicates the transition between one-way steerability and both-way steerability for the two-mode Gaussian state under the influence of Hawking radiation.

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