Einstein-Podolsky-Rosen steering and the steering ellipsoid

TL;DR

This paper investigates the steerability of two-qubit states using the quantum steering ellipsoid formalism, providing necessary and sufficient conditions for EPR-steering, and evaluates related integrals over hemispheres.

Contribution

It introduces new necessary and sufficient conditions for two-qubit steerability based on the steering ellipsoid and local hidden state models.

Findings

Derived a strong necessary condition for steerability.

Established two provably sufficient conditions via asymmetric EPR-steering inequalities.

Evaluated integrals over hemispheres for positive matrices in the context of quantum steering.

Abstract

The question of which two-qubit states are steerable (i.e. permit a demonstration of EPR-steering) remains open. Here, a strong necessary condition is obtained for the steerability of two-qubit states having maximally-mixed reduced states, via the construction of local hidden state models. It is conjectured that this condition is in fact sufficient. Two provably sufficient conditions are also obtained, via asymmetric EPR-steering inequalities. Our work uses ideas from the quantum steering ellipsoid formalism, and explicitly evaluates the integral of $\boldsymbol n/(\boldsymbol n^\intercal A\boldsymbol n)^2$ over arbitrary unit hemispheres for any positive matrix $A$.