# One-Shot Randomized and Nonrandomized Partial Decoupling

## TL;DR

This paper introduces a new quantum information task called partial decoupling, providing bounds on state closeness after applying random unitaries and channels, generalizing the one-shot decoupling theorem.

## Contribution

It generalizes the one-shot decoupling theorem to partial decoupling with a direct-sum-product decomposition, offering bounds based on smooth conditional entropies.

## Key findings

- Derived bounds on the average distance between states
- Provided a generalization of the one-shot decoupling theorem
- Applicable to quantum information theory and physics

## Abstract

We introduce a task that we call partial decoupling, in which a bipartite quantum state is transformed by a unitary operation on one of the two subsystems and then is subject to the action of a quantum channel. We assume that the subsystem is decomposed into a direct-sum-product form, which often appears in the context of quantum information theory. The unitary is chosen at random from the set of unitaries having a simple form under the decomposition. The goal of the task is to make the final state, for typical choices of the unitary, close to the averaged final state over the unitaries. We consider a one-shot scenario, and derive upper and lower bounds on the average distance between the two states. The bounds are represented simply in terms of smooth conditional entropies of quantum states involving the initial state, the channel and the decomposition. Thereby we provide generalizations of the one-shot decoupling theorem. The obtained result would lead to further development of the decoupling approaches in quantum information theory and fundamental physics.