The resource theory of steering

TL;DR

This paper develops an operational resource theory framework for Einstein-Podolsky-Rosen steering, introducing new quantifiers and conditions for state conversions, with implications for understanding quantum non-locality.

Contribution

It introduces convex steering monotones, characterizes steering non-increasing operations, and establishes conditions for pure-state steering conversions and the non-existence of steering bits.

Findings

Relative entropy of steering as a quantifier

Steerable weight and robustness are convex steering monotones

No measure-independent maximally steerable states exist

Abstract

We present an operational framework for Einstein-Podolsky-Rosen steering as a physical resource. To begin with, we characterize the set of steering non-increasing operations (SNIOs) --i.e., those that do not create steering-- on arbitrary-dimensional bipartite systems composed of a quantum subsystem and a black-box device. Next, we introduce the notion of convex steering monotones as the fundamental axiomatic quantifiers of steering. As a convenient example thereof, we present the relative entropy of steering. In addition, we prove that two previously proposed quantifiers, the steerable weight and the robustness of steering, are also convex steering monotones. To end up with, for minimal-dimensional systems, we establish, on the one hand, necessary and sufficient conditions for pure-state steering conversions under stochastic SNIOs and prove, on the other hand, the non-existence of steering bits, i.e., measure-independent maximally steerable states from which all states can be obtained by means of the free operations. Our findings reveal unexpected aspects of steering and lay foundations for further resource-theory approaches, with potential implications in Bell non-locality.