Quantum Steering Ellipsoids

TL;DR

This paper introduces a geometric representation called the quantum steering ellipsoid for two-qubit states, revealing new features like incomplete steering in separable states and linking entanglement to geometric properties.

Contribution

It provides an elementary construction of the ellipsoid, calculates its volume, and establishes a geometric criterion for separability based on a nested tetrahedron condition.

Findings

Existence of incomplete steering in separable states

Entanglement characterized by three geometric features

Separable states obey a nested tetrahedron condition

Abstract

The quantum steering ellipsoid of a two-qubit state is the set of Bloch vectors that Bob can collapse Alice's qubit to, considering all possible measurements on his qubit. We provide an elementary construction of the ellipsoid for arbitrary states, calculate its volume, and explain how this geometric representation can be made faithful. The representation provides a range of new results, and uncovers new features, such as the existence of "incomplete steering" in separable states. We show that entanglement can be analysed in terms of three geometric features of the ellipsoid, and prove that a state is separable if and only if it obeys a "nested tetrahedron" condition.