We study the dynamics of Gaussian quantum steering in the background of a Garfinkle-Horowitz-Strominger dilaton black hole. It is found that the gravity induced by dilaton field will destroy the quantum steerability between the inertial observer Alice and the observer Bob who hovers outside the event horizon, while it generates steering-type quantum correlations between the causally disconnected regions. Therefore, the observers can steer each others state by local measurements even though they are separated by the event horizon. Unlike quantum entanglement in the dilaton spacetime, the quantum steering experiences catastrophic behaviors such as sudden death and sudden birth with increasing dilaton charge. In addition, the dilaton gravity destroys the symmetry of Gaussian steering and the latter is always asymmetric in the dilation spacetime. Interestingly, the attainment of maximal steering asymmetry indicates the critical point between one-way and two-way steering for the two-mode Gaussian state in the dilaton spacetime.
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