Scaling of Einstein-Podolsky-Rosen steering in spin chains

Symmetric quantum properties and correlations have been often used earlier to study quantum phase transitions in many body systems and spin models. However, the use of asymmetric quantum features, such as steering, have attracted smaller amount of attention in this context, so far. We study EPR steering and quantum phase transitions in the Ising model in transverse field and in the anisotropic XY model by using steering robustness and quantum renormalization group method. The key ingredients of the quantum criticality near the critical points, such as finite-size scaling behaviour and critical exponents, are investigated in detail with two commonly used spin models. Our results show that the first derivative of steering robustness between two blocks diverges near the quantum phase transition points for both models, and exhibits a finite-size scaling effect. Moreover, we explore in detail the asymmetric character of EPR steering, by taking into accounts finite-size effects and measurement number, in the reduced block state in the anisotropic XY model. The results imply that one-way EPR steering does not exist in large system under the limit L ->infinity despite the fact that EPR steering for the reduced block state is asymmetric.