# The Duality in Quantum Work

## TL;DR

This paper introduces a wave-particle duality perspective to quantum work statistics, defining predictability and coherence effectiveness, and establishing a fundamental tradeoff relation that advances understanding of quantum coherence effects in thermodynamics.

## Contribution

It proposes a novel duality framework for quantum work, defining predictability and coherence measures that obey a fundamental inequality, enhancing the theoretical understanding of quantum coherence in thermodynamics.

## Key findings

- Predictability and coherence effectiveness satisfy a fundamental inequality.
- The tradeoff relation between population and coherence contributions is established.
- Application to a driven two-level system illustrates the bound conditions.

## Abstract

An open question of fundamental importance in quantum thermodynamics is how to describe the statistics of work for initial state with quantum coherence. In this paper, work statistics is considered from a fully new perspective of "wave-particle" duality. Based on the generalized quantum work measurement, predictability of energy levels $\mathcal{D}_W$ and effectiveness of coherence $\mathcal{V}_W$ are defined, and they obey inequality $\mathcal{D}_W^2+\mathcal{V}_W^2\leq1$, which is the fundamental tradeoff relations between the contributions of population and coherence to quantum work distribution. As an application, we consider a driven two-level system and discuss the condition of the bound of above tradeoff relation. These results shed light on the effects of quantum coherence in quantum thermodynamics.