# Leggett-Garg Inequalities for Quantum Fluctuating Work

^{*}

## Abstract

**:**

## 1. Introduction

- Macrorealism per se: physical observables take on well-defined values at all times independent of the act of observation.
- Non-invasive measurability: in principle it is possible to measure the value of an observable without changing the subsequent evolution of the system.

## 2. Inequalities for Moments of Work

## 3. Violations of the Leggett-Garg Inequalities for Work Moments

## 4. Inequalities for the Characteristic Function of Work

## 5. Generalisation to Weak Measurements of Work

## 6. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Derivation of Equation (5)

## Appendix B. Derivation of Equation (8)

## Appendix C. Derivation of Equations (13) and (14)

**Table A1.**Displays the various combinations of ${\theta}_{01}$ and ${\theta}_{12}$ in Equation (A14).

${\theta}_{01}$ | ${\theta}_{12}$ | $\mathrm{cos}\left({\theta}_{12}\right)+\mathrm{cos}\left({\theta}_{01}\right)-\mathrm{cos}({\theta}_{12}+{\theta}_{01})$ | $\mathrm{sin}\left({\theta}_{12}\right)+\mathrm{sin}\left({\theta}_{01}\right)-\mathrm{sin}({\theta}_{12}+{\theta}_{01})$ |
---|---|---|---|

0 | 0 | 1 | 0 |

0 | $-\lambda \u03f5$ | 1 | 0 |

0 | $\lambda \u03f5$ | 1 | 0 |

$-\lambda \u03f5$ | $+\lambda \u03f5$ | $2\mathrm{cos}\left(\lambda \u03f5\right)-1$ | 0 |

$\lambda \u03f5$ | $-\lambda \u03f5$ | $2\mathrm{cos}\left(\lambda \u03f5\right)-1$ | 0 |

$-\lambda \u03f5$ | 0 | 1 | 0 |

$\lambda \u03f5$ | 0 | 1 | 0 |

## Appendix D. Maximal Violations for the Characteristic Function

## Appendix E. Maximum Violations for Full-Counting Statistics

## Appendix F. Maximum Violations for the Margenau-Hill Distribution

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**Figure 1.**Schematic diagram for detecting non-classical work statistics. An observer performs three separate experiments, (

**a**–

**c**), in which the fluctuating work done on the system is measured between the time intervals shown in the diagram. To test the validity of the Leggett-Garg inequality for work, Equation (5), one compares the statistics of the three experiments, with the same initial state chosen at time ${t}_{0}$. Note that in experiment (

**c**) no measurement is made at ${t}_{0}$, thus the system evolves unitarily up to time ${t}_{1}$.

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Miller, H.J.D.; Anders, J. Leggett-Garg Inequalities for Quantum Fluctuating Work. *Entropy* **2018**, *20*, 200.
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Miller HJD, Anders J. Leggett-Garg Inequalities for Quantum Fluctuating Work. *Entropy*. 2018; 20(3):200.
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**Chicago/Turabian Style**

Miller, Harry J. D., and Janet Anders. 2018. "Leggett-Garg Inequalities for Quantum Fluctuating Work" *Entropy* 20, no. 3: 200.
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