# Quantum coherence between subspaces: State transformation, Cohering Power, $k$-coherence and other properties

## TL;DR

This paper develops a framework for block-coherence in quantum systems, analyzing state transformations, coherence powers of channels, and the relation to $k$-coherence, advancing understanding of quantum resource theories.

## Contribution

It introduces a new framework for block-coherence, characterizes state transformations via majorization, and explores coherence powers and their relation to $k$-coherence.

## Key findings

- Majorization condition is necessary and sufficient for state transformation.
- Identified the form of maximally coherent states for resource construction.
- Determined block-cohering and decohering powers for various quantum channels.

## Abstract

The concept of bock-coherence, first introduced in [1] and developed in [2,3] encompasses the case where experimental capabilities are not so delicate to perform arbitrary refined measurements on individual atoms. We develop a framework which facilitates further investigation of this resource theory in several respects. Using this framework, we investigate the problem of state conversion by incoherent operations and show that a majorization condition is the necessary and sufficient condition for state transformation by block-incoherent operations. We also determine the form of the maximally coherent state from which all other states and all unitary gates can be constructed by incoherent operations. Thereafter, we define the concept of block-cohering and block-decohering powers of quantum channels and determine these powers for several types of channels. Finally, we explore the relation between block coherence and a previous extension of coherence, known as $k$-coherence.