The logarithmic negativity: A full entanglement monotone that is not convex

TL;DR

It is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation (a set of operations that incorporate local operations and classical communication as a subset), which is surprising, as it is generally considered that convexity describes the local physical process of losing information.

Abstract

It is proven that the logarithmic negativity does not increase on average under positive partial transpose preserving (PPT) operation including subselection (a set of operations that incorporate local operations and classical communication (LOCC) as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is obtained from the negativity applying a concave function and is itself not a convex quantity this result is surprising as convexity is generally considered as describing the local physical process of losing information. The role of convexity and in particular its relation (or lack thereof) to physical processes is discussed in this context.

Figures (1)

• Figure 1: Schematic picture of the action of quantum operations with and without sub-selection shown in part (a) and part (b) respectively.