# Asymmetry and coherence weight of quantum states

## TL;DR

This paper introduces the concept of asymmetry weight as a resource measure in quantum states, explores its properties, connections to entanglement and coherence, and provides exact formulas for Werner states.

## Contribution

It defines asymmetry weight and coherence weight, analyzes their properties, and links them to other quantum resource measures, revealing new operational interpretations and exact results for Werner states.

## Key findings

- Asymmetry weight is a convex, monotonic resource measure.
- Asymmetry weight can be viewed as a state-dependent asymmetry witness.
- For Werner states, coherence weight, robustness, and l1 norm are equal.

## Abstract

The asymmetry of quantum states is an important resource in quantum information processing tasks such as quantum metrology and quantum communication. In this paper, we introduce the notion of $asymmetry~weight$ --- an operationally motivated asymmetry quantifier in the resource theory of asymmetry. We study the convexity and monotonicity properties of asymmetry weight and focus on its interplay with the corresponding semidefinite programming (SDP) forms along with its connection to other asymmetry measures. Since the SDP form of asymmetry weight is closely related to asymmetry witnesses, we find that the asymmetry weight can be regarded as a (state-dependent) asymmetry witness. Moreover, some specific entanglement witnesses can be viewed as a special case of an asymmetry witness --- which indicates a potential connection between asymmetry and entanglement. We also provide an operationally meaningful coherence measure, which we term $coherence~weight$, and investigate its relationship to other coherence measures like the robustness of coherence and the $l_1$ norm of coherence. In particular, we show that for Werner states in any dimension $d$, all three coherence quantifiers, namely, the coherence weight, the robustness of coherence, and the $l_1$ norm of coherence, are equal and are given by a single letter formula.