Peres-type Criterion of Einstein-Podolsky-Rosen Steering for Two Qubits

Quantum nonlocality manifests in multipartite systems through entanglement, Bell's nonlocality, and Einstein-Podolsky-Rosen (EPR) steering. While Peres's positive-partial-transpose criterion provides a simple and powerful test for entanglement, a comparably elegant spectral criterion for detecting EPR steering remains an open challenge. In this work, we systematically explore whether a Peres-type criterion can be established for EPR steering in the two-qubit system. Focusing on rank-2 (including rank-1) states and the two-qubit Werner state, we analyze the eigenvalues of their partially transposed density matrices and construct a significant steering criterion based on symmetric combinations of these eigenvalues. We prove that this criterion serves as a necessary and sufficient condition for steerability for the Werner state, all two-qubit pure states, all two-qubit rank-2 states. Furthermore, we validate the criterion for higher-rank states (rank-3 and rank-4) and show that the results align with known steering inequalities. Our findings suggest a more unified framework for detecting quantum nonlocality via partial transposition and open avenues for further theoretical and numerical investigations into steering detection.