# Quantifying coherence with quantum addition

## TL;DR

This paper introduces a new way to quantify quantum coherence using quantum addition channels, deriving inequalities and bounds that deepen understanding of coherence in mixed quantum states.

## Contribution

It proves a reverse entropy power equality and establishes a new coherence measure based on quantum addition channels, with applications to uncertainty relations.

## Key findings

- Quantitative measure of quantum coherence from quantum addition channels
- Proved a reverse entropy power equality for quantum systems
- Formulated a state-dependent uncertainty relation for two observables

## Abstract

Quantum addition channels have been recently introduced in the context of deriving entropic power inequalities for finite dimensional quantum systems. We prove a reverse entropy power equality which can be used to analytically prove an inequality conjectured recently for arbitrary dimension and arbitrary addition weight. We show that the relative entropic difference between the output of such a quantum additon channel and the corresponding classical mixture quantitatively captures the amount of coherence present in a quantum system. This new coherence measure admits an upper bound in terms of the relative entropy of coherence and is utilized to formulate a state-dependent uncertainty relation for two observables. Our results may provide deep insights to the origin of quantum coherence for mixed states that truly come from the discrepancy between quantum addition and the classical mixture.