# Quantitative Coherence Witness for Finite Dimensional States

## TL;DR

This paper introduces a quantitative coherence witness for finite-dimensional quantum states, enabling direct measurement of coherence and providing an operational interpretation of the $l_1$-norm of coherence.

## Contribution

The paper proves the existence of stringent coherence witnesses for all finite-dimensional states and links their mean values to the $l_1$-norm of coherence, offering a practical measurement method.

## Key findings

- Existence of coherence witnesses for all finite-dimensional states.
- Mean value of the witness bounds the $l_1$-norm of coherence.
- Operational interpretation of the $l_1$-norm of coherence.

## Abstract

We define the stringent coherence witness as an observable which has zero mean value for all of incoherent states and hence a nonzero mean value indicates the coherence. The existence of such witnesses are proved for any finite-dimension states. Not only is the witness efficient in testing whether the state is coherent, the mean value is also quantitatively related to the amount of coherence contained in the state. For an unknown state, the modulus of the mean value of a normalized witness provides a tight lower bound of $l_1$-norm of coherence in the state. When we have some previous knowledge of the state, the optimal witness is derived such that its measured mean value, called the witnessed coherence, equals to the $l_1$-norm of coherence. One can also fix the witness and implement some incoherent operations on the state before the witness is measured. In this case, the measured mean value cannot reach the witnessed coherence if the initial state and the fixed witness does not match well. Based this result, we design a quantum coherence game. Our results provides a way to directly measure the coherence in arbitrary finite dimension states and an operational interpretation of the $l_1$-norm of coherence.