# Coherence number as a discrete quantum resource

## TL;DR

The paper introduces the coherence number, a new discrete measure of quantum coherence, and explores its implications for entanglement conversion and the Grover search algorithm, revealing its role as an optimal resource.

## Contribution

It defines the coherence number as a new monotone, establishes its relation to entanglement conversion, and analyzes its behavior in Grover's algorithm.

## Key findings

- Coherence number generalizes coherence rank to mixed states.
- A necessary and sufficient condition links coherence number to entanglement conversion.
- Coherence number drops abruptly at maximal success probability in Grover's search.

## Abstract

We introduce a new discrete coherence monotone named the \emph{coherence number}, which is a generalization of the coherence rank to mixed states. After defining the coherence number in a similar manner to the Schmidt number in entanglement theory, we present a necessary and sufficient condition of the coherence number for a coherent state to be converted to an entangled state of nonzero $k$-concurrence (a member of the generalized concurrence family with $2\le k \le d$). It also turns out that the coherence number is a useful measure to understand the process of Grover search algorithm of $N$ items. We show that the coherence number remains $N$ and falls abruptly when the success probability of the searching process becomes maximal. This phenomenon motivates us to analyze the depletion pattern of $C_c^{(N)}$ (the last member of the generalized coherence concurrence, nonzero when the coherence number is $N$), which turns out to be an optimal resource for the process since it is completely consumed to finish the searching task.