Relative entropy of steering: On its definition and properties

TL;DR

This paper proposes a new definition for the relative entropy of steering rooted in quantum Shannon theory, proving its convexity, monotonicity, continuity, and faithfulness, and explores a restricted version relevant for practical scenarios.

Contribution

It introduces a modified, well-grounded definition of the relative entropy of steering and establishes its key properties, including convexity, monotonicity, continuity, and faithfulness, along with a restricted variant.

Findings

The modified relative entropy of steering is a convex steering monotone.

It is uniformly continuous and faithful, with quantitative bounds.

A restricted version is also convex, monotone, continuous, and faithful.

Abstract

In [Gallego and Aolita, Physical Review X 5, 041008 (2015)], the authors proposed a definition for the relative entropy of steering and showed that the resulting quantity is a convex steering monotone. Here we advocate for a different definition for relative entropy of steering, based on well grounded concerns coming from quantum Shannon theory. We prove that this modified relative entropy of steering is a convex steering monotone. Furthermore, we establish that it is uniformly continuous and faithful, in both cases giving quantitative bounds that should be useful in applications. We also consider a restricted relative entropy of steering which is relevant for the case in which the free operations in the resource theory of steering have a more restricted form (the restricted operations could be more relevant in practical scenarios). The restricted relative entropy of steering is convex, monotone with respect to these restricted operations, uniformly continuous, and faithful.