# Additivity of entanglement of formation via entanglement-breaking space

## TL;DR

This paper investigates the structure of entanglement-breaking spaces to understand when the entanglement of formation is additive, providing new constructions and characterizations in specific Hilbert spaces and analyzing related entanglement costs.

## Contribution

It systematically constructs and characterizes entanglement-breaking spaces in certain bipartite systems, revealing conditions for additivity of entanglement of formation and extending previous examples.

## Key findings

- Constructed EB spaces in $bC^m\otimes\bbC^3$ and $bC^2\otimes\bbC^n$.
- Characterized two-qubit states of rank two with nonadditive EOF.
- Connected results to known examples and analyzed entanglement cost.

## Abstract

We study the entanglement-breaking (EB) space, such that the entanglement of formation (EOF) of a bipartite quantum state is additive when its range is an EB subspace. We systematically construct the EB spaces in the Hilbert space $\bbC^m\otimes\bbC^3$, and the $2$-dimensional EB space in $\bbC^2\otimes\bbC^n$. We characterize the expression of two-qubit states of rank two with nonadditive EOF, if they exist. We further apply our results to construct EB spaces of an arbitrarily given dimensions. We show that the example in [PRL 89,027901(2002)] is a special case of our results. We further work out the entanglement cost of a qubit-qutrit state in terms of the two-atom system of the .