# $\alpha$-Logarithmic negativity

## TL;DR

This paper introduces a family of $oldsymbol{ extalpha}$-logarithmic negativity measures that interpolate between logarithmic negativity and $oldsymbol{ extkappa}$-entanglement, providing a unified framework with proven properties and extensions to channels and resource theories.

## Contribution

It establishes a continuum of entanglement measures linking logarithmic negativity and $oldsymbol{ extkappa}$-entanglement, with proofs of key properties and generalizations to channels and resource theories.

## Key findings

- $oldsymbol{ extalpha}$-logarithmic negativity is an entanglement monotone.
- It satisfies normalization, faithfulness, and subadditivity.
- It is neither convex nor monogamous.

## Abstract

The logarithmic negativity of a bipartite quantum state is a widely employed entanglement measure in quantum information theory, due to the fact that it is easy to compute and serves as an upper bound on distillable entanglement. More recently, the $\kappa$-entanglement of a bipartite state was shown to be the first entanglement measure that is both easily computable and has a precise information-theoretic meaning, being equal to the exact entanglement cost of a bipartite quantum state when the free operations are those that completely preserve the positivity of the partial transpose [Wang and Wilde, Phys. Rev. Lett. 125(4):040502, July 2020]. In this paper, we provide a non-trivial link between these two entanglement measures, by showing that they are the extremes of an ordered family of $\alpha$-logarithmic negativity entanglement measures, each of which is identified by a parameter $\alpha\in[ 1,\infty] $. In this family, the original logarithmic negativity is recovered as the smallest with $\alpha=1$, and the $\kappa$-entanglement is recovered as the largest with $\alpha=\infty$. We prove that the $\alpha $-logarithmic negativity satisfies the following properties: entanglement monotone, normalization, faithfulness, and subadditivity. We also prove that it is neither convex nor monogamous. Finally, we define the $\alpha$-logarithmic negativity of a quantum channel as a generalization of the notion for quantum states, and we show how to generalize many of the concepts to arbitrary resource theories.