Gaussian tripartite steering in Schwarzschild black hole

Multipartite steering is a fundamental quantum resource that is uniquely suited to tackling complex relativistic quantum information challenges, but its properties in the gravitational field context remain to be elucidated. We study the distribution of Gaussian tripartite steering in the background of a Schwarzschild black hole. Our results show that physically accessible $2\rightarrow 1$ steerability remains robust at any Hawking temperature, which contrasts with the behavior of $1\rightarrow 1$ steering that experiences ``sudden death" as the Hawking temperature increases, making $2\rightarrow 1$ steerability a more reliable candidate for relativistic quantum information tasks. Notably, we observe that the sudden death of quantum steering is accompanied by a peak in steering asymmetry, signifying a critical transition from two-way steering to one-way steering in the relativistic three-mode system. We find that the Hawking effect can generate inaccessible two-way steering of a bipartite system in both directions, as shown in [Phys. Rev. D 93, 125011 (2016)], while it can only generate physically inaccessible one-way $2\rightarrow 1$ steering in tripartite systems. These insights enhance our understanding of the redistribution of multipartite steering near the event horizon of the black hole.