Epsilon-measures of entanglement

TL;DR

This paper introduces epsilon-measures of entanglement, which are parameter-dependent variants of traditional measures designed for realistic scenarios with partial information, inheriting many properties and offering continuity where original measures may lack.

Contribution

It defines epsilon-measures of entanglement, demonstrating their properties, inheritance from original measures, and their role as smoothed, continuous variants.

Findings

Epsilon-measures inherit weak monotonicity under LOCC.

They may increase on average under stochastic LOCC.

Epsilon-version of convex measures is continuous even if original is not.

Abstract

We associate to every entanglement measure a family of measures which depend on a precision parameter, and which we call epsilon-measures of entanglement. Their definition aims at addressing a realistic scenario in which we need to estimate the amount of entanglement in a state that is only partially known. We show that many properties of the original measure are inherited by the family, in particular weak monotonicity under transformations applied by means of Local Operations and Classical Communication (LOCC). On the other hand, they may increase on average under stochastic LOCC. Remarkably, the epsilon-version of a convex entanglement measure is continuous even if the original entanglement measure is not, so that the epsilon-version of an entanglement measure may be actually considered a smoothed version of it.