# One-shot entanglement distillation beyond local operations and classical communication

## TL;DR

This paper investigates one-shot entanglement distillation under various quantum operation classes, establishing formal relationships with entanglement measures, and providing exact characterizations and computational methods for different state classes.

## Contribution

It introduces a formal framework for comparing distillation performance across operation classes and characterizes the one-shot distillable entanglement for several state types.

## Key findings

- One-shot distillable entanglement of pure states is identical across multiple operation classes.
- Distillation rates for isotropic and maximally correlated states can be computed via linear or semidefinite programs.
- All operation classes perform equally in environment-assisted entanglement distillation.

## Abstract

We study the task of entanglement distillation in the one-shot setting under different classes of quantum operations which extend the set of local operations and classical communication (LOCC). Establishing a general formalism which allows for a straightforward comparison of their exact achievable performance, we relate the fidelity of distillation under these classes of operations with a family of entanglement monotones and the rates of distillation with a class of smoothed entropic quantities based on the hypothesis testing relative entropy. We then characterise exactly the one-shot distillable entanglement of several classes of quantum states and reveal many simplifications in their manipulation. We show in particular that the $\varepsilon$-error one-shot distillable entanglement of any pure state is the same under all sets of operations ranging from one-way LOCC to separability-preserving operations or operations preserving the set of states with positive partial transpose, and can be computed exactly as a quadratically constrained linear program. We establish similar operational equivalences in the distillation of isotropic and maximally correlated states, reducing the computation of the relevant quantities to linear or semidefinite programs. We also show that all considered sets of operations achieve the same performance in environment-assisted entanglement distillation from any state.