# Useful states and entanglement distillation

## TL;DR

This paper establishes upper bounds on the distillable entanglement of mixed quantum states using convex decompositions into useful and useless states, with applications to quantum channels and specific state classes.

## Contribution

It introduces a convex decomposition framework for bounding distillable entanglement, connecting useful states to degradable and maximally correlated states, and extends the method to quantum channels.

## Key findings

- Upper bounds are derived based on convex decompositions.
- The bounds are tighter than previous bounds for certain channels.
- The method applies to isotropic and Werner states, simplifying optimization.

## Abstract

We derive general upper bounds on the distillable entanglement of a mixed state under one-way and two-way LOCC. In both cases, the upper bound is based on a convex decomposition of the state into 'useful' and 'useless' quantum states. By 'useful', we mean a state whose distillable entanglement is non-negative and equal to its coherent information (and thus given by a single-letter, tractable formula). On the other hand, 'useless' states are undistillable, i.e., their distillable entanglement is zero. We prove that in both settings the distillable entanglement is convex on such decompositions. Hence, an upper bound on the distillable entanglement is obtained from the contributions of the useful states alone, being equal to the convex combination of their coherent informations. Optimizing over all such decompositions of the input state yields our upper bound. The useful and useless states are given by degradable and antidegradable states in the one-way LOCC setting, and by maximally correlated and PPT states in the two-way LOCC setting, respectively. We also illustrate how our method can be extended to quantum channels. Interpreting our upper bound as a convex roof extension, we show that it reduces to a particularly simple, non-convex optimization problem for the classes of isotropic states and Werner states. In the one-way LOCC setting, this non-convex optimization yields an upper bound on the quantum capacity of the qubit depolarizing channel that is strictly tighter than previously known bounds for large values of the depolarizing parameter. In the two-way LOCC setting, the non-convex optimization achieves the PPT-relative entropy of entanglement for both isotropic and Werner states.