Einstein-Podolsky-Rosen steering: Its geometric quantification and witness
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.
Keywords
Cite
@article{arxiv.1709.06703,
title = {Einstein-Podolsky-Rosen steering: Its geometric quantification and witness},
author = {Huan-Yu Ku and Shin-Liang Chen and Costantino Budroni and Adam Miranowicz and Yueh-Nan Chen and Franco Nori},
journal= {arXiv preprint arXiv:1709.06703},
year = {2018}
}