Einstein-Podolsky-Rosen steering: Its geometric quantification and witness

TL;DR

This paper introduces a geometric measure of quantum steerability based on trace distance, with methods to estimate it via semidefinite programming, and visual tools like LHS surfaces and steering ellipsoids for qubit states.

Contribution

It proposes a convex steering monotone and a steerability witness using geometric and optimization techniques, enhancing understanding and detection of quantum steerability.

Findings

Defined a trace-distance-based steerability measure.

Developed bounds and geometric interpretations for the measure.

Provided visualization tools for qubit steerability properties.

Abstract

We propose a measure of quantum steerability, namely a convex steering monotone, based on the trace distance between a given assemblage and its corresponding closest assemblage admitting a local-hidden-state (LHS) model. We provide methods to estimate such a quantity, via lower and upper bounds, based on semidefinite programming. One of these upper bounds has a clear geometrical interpretation as a linear function of rescaled Euclidean distances in the Bloch sphere between the normalized quantum states of: (i) a given assemblage and (ii) an LHS assemblage. For a qubit-qubit quantum state, the above ideas also allow us to visualize various steerability properties of the state in the Bloch sphere via the so-called LHS surface. In particular, some steerability properties can be obtained by comparing such an LHS surface with a corresponding quantum steering ellipsoid. Thus, we propose a witness of steerability corresponding to the difference of the volumes enclosed by these two surfaces. This witness (which reveals the steerability of a quantum state) enables finding an optimal measurement basis, which can then be used to determine the proposed steering monotone (which describes the steerability of an assemblage) optimized over all mutually-unbiased bases.