Non-separability and steerability of two-qubit states from the geometry of steering outcomes

TL;DR

This paper reveals that the 4-dimensional geometry of Alice's steering outcomes fully characterizes the non-separability and steerability of two-qubit states, providing a geometric classification framework.

Contribution

It introduces a geometric approach using 4-dimensional skewed double-cones to classify two-qubit states by their non-separability and steerability.

Findings

The geometry of steering outcomes determines non-separability.

Steerability classification reduces to geometric classification.

Provides a new geometric perspective on quantum steering.

Abstract

When two qubits A and B are in an appropriate state, Alice can remotely steer Bob's system B into different ensembles by making different measurements on A. This famous phenomenon is known as quantum steering, or Einstein-Podolsky-Rosen steering. Importantly, quantum steering establishes the correspondence not only between a measurement on A (made by Alice) and an ensemble of B (owned by Bob) but also between each of Alice's measurement outcomes and an unnormalized conditional state of Bob's system. The unnormalized conditional states of B corresponding to all possible measurement outcomes of Alice are called Alice's steering outcomes. We show that, surprisingly, the $4$-dimensional geometry of Alice's steering outcomes completely determines both the non-separability of the two-qubit state and its steerability from her side. Consequently, the problem of classifying two-qubit states into non-separable and steerable classes is equivalent to geometrically classifying certain $4$-dimensional skewed double-cones.