Analytical Steering Criterion Based on Critical Radius for any Two‐Qubit States

ABSTRACT Quantum steering refers to the phenomenon where one quantum system can instantaneously influence another via local measurements. The critical radius, proposed by Nguyen et al. [Phys. Rev. Lett. 122, 240401 (2019)], is a sufficient‐and‐necessary steering criterion (SNSC). However, its broad applicability has been restricted by the absence of analytical solutions for general two‐qubit states. To address this, we develop the first comprehensive analytical framework for the critical radius, unifying theory and practice. We first derive an exact SNSC for all two‐qubit T‐states under infinite measurements and verify its equivalence to the critical radius. Leveraging the critical radius's symmetry and concavity, we generalize the framework to arbitrary two‐qubit states, aiming to transform the upper and lower bounds of the critical radius into an equal pattern via optimized local operations, thereby deriving an accurate formula for the critical radius. Critically, we discover the first exact analytical SNSCs for two distinct classes of asymmetric states, precisely mapping the boundary between steerable and unsteerable regions. These breakthroughs eliminate reliance on the numerical semidefinite programming methods, provide an operationally efficient method for discovering the hidden steerability.