Hawking effect on the universal relationship between quantum steering and the first-order coherence in Schwarzschild spacetime

Quantum steering and coherence are fundamental resources for quantum computation and quantum communication. Understanding their interplay under different environments is crucial for optimizing quantum information tasks. In this letter, we derive a complementary relation between the steering criterion based on infinite measurements and the first-order coherence. Then, we research Hawking effect on our complementarity formula in Schwarzschild spacetime. Quantum steering always decreases as the Hawking temperature increases, but the complementary relation between quantum steering and the first-order coherence for two-qubit states is satisfied, which means that quantum resources have been redistributed in this case. Our derived equation offers the convenient analytical solution of the steering criterion based on infinite measurements for arbitrary two-qubit states only using the first-order coherence and the purity of the state, thereby opening up new possibilities for further research on quantum steering. This research not only deepens our understanding on the relationship between quantum steering and coherence, but also provides a more accessible method for quantifying steering in two-qubit systems-a significant simplification for experimental implementations.