One pure steered state implies Einstein-Podolsky-Rosen steering

TL;DR

This paper demonstrates that the presence of at least one pure steered state in a two-qubit entangled system guarantees two-way EPR steering, extending the understanding of quantum steering and its geometric properties.

Contribution

It proves that a single pure steered state implies two-way EPR steering in two-qubit entangled states, linking geometric properties of steering ellipsoids to steerability.

Findings

Pure steered states imply EPR steerability from Alice to Bob.

The number of tangent points of steering ellipsoids relates to steerability.

Single pure steered state guarantees two-way EPR steering.

Abstract

In this work, we show that a two-qubit entangled state admitting at least one pure steered state is Einstein-Podolsky-Rosen (EPR) steerable from Alice to Bob. Pure steered states signifies that the quantum steering ellipsoid of Bob is tangent to his Bloch sphere at least at a single point. Furthermore, we prove that for a two-qubit entangled state, Bob's quantum steering ellipsoid is tangent to his Bloch sphere at exactly $N$ points, for $N\in \{ 0,1,2,\infty\}$, if and only if Alice's quantum steering ellipsoid is tangent to her Bloch sphere at exactly $N$ points. For any two-qubit entangled state, therefore, if one party can steer the other to at least one pure state, the state is two-way EPR steerable. We also present several illuminating instances of two-qubit entangled states such that the EPR steering can be verified in terms of pure steered states. Our result addresses the Gisin theorem in a EPR steering scenario: at least a single pure steered state implies two-way steering.