# k-stretchability of entanglement, and the duality of k-separability and k-producibility

## TL;DR

This paper explores the duality between k-separability and k-producibility in multipartite entanglement, introducing the concept of k-stretchability and a classification scheme based on Young diagrams to analyze entanglement and correlation properties.

## Contribution

It reveals a partial duality between k-separability and k-producibility, and introduces the new concept of k-stretchability within a unified classification framework.

## Key findings

- Identifies a duality between k-separability and k-producibility.
- Introduces the concept of k-stretchability as a balanced measure.
- Develops a classification scheme using Young diagrams for entanglement properties.

## Abstract

The notions of $k$-separability and $k$-producibility are useful and expressive tools for the characterization of entanglement in multipartite quantum systems, when a more detailed analysis would be infeasible or simply needless. In this work we reveal a partial duality between them, which is valid also for their correlation counterparts. This duality can be seen from a much wider perspective, when we consider the entanglement and correlation properties which are invariant under the permutations of the subsystems. These properties are labeled by Young diagrams, which we endow with a refinement-like partial order, to build up their classification scheme. This general treatment reveals a new property, which we call $k$-stretchability, being sensitive in a balanced way to both the maximal size of correlated (or entangled) subsystems and the minimal number of subsystems uncorrelated with (or separable from) one another.