Classifying quantum steering, entanglement, and discord for continuous variables in Schwarzschild spacetime

Quantum steering, entanglement, and discord often deal with different relativistic quantum information tasks. It is necessary, therefore, to classify different hierarchies of quantum correlations for continuous variables in Schwarzschild spacetime. We obtain the conditions for three cases: (i) quantum steering, entanglement, and discord survive; (ii) quantum entanglement is not steerable; and (iii) quantum discord is not entangled and steerable in Schwarzschild spacetime. It means that quantum steering and entanglement suffer from a "sudden death" with the growth of the Hawking temperature, while quantum discord vanishes only at the infinite Hawking temperature. In addition, the maximal steering asymmetry indicates the transition between one-way steering and two-way steering. These results may have important implications for guiding relativistic quantum information tasks.