# Daemonic Ergotropy: Generalised Measurements and Multipartite Settings

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## Abstract

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**3**, 12 (2018)]. We prove that quantum correlations are not advantageous over classical correlations if projective measurements are considered. We go beyond the limitations of the original definition to include generalised measurements and provide an example in which this allows for a higher daemonic ergotropy. Moreover, we propose a see-saw algorithm to find a measurement that attains the maximum work extraction. Finally, we provide a multipartite generalisation of daemonic ergotropy that pinpoints the influence of multipartite quantum correlations, and study it for multipartite entangled and classical states.

## 1. Introduction

## 2. Notation and Concepts

## 3. Non-Optimality of Projective Measurements for Daemonic Ergotropy

## 4. Construction of Optimal POVMs

**Lemma**

**1.**

**Proof.**

**Corollary**

**2.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

Algorithm 1 Optimise POVM for daemonic ergotropy | |

1: Choose n different unitaries ${U}_{i}$ and calculate ${\tau}_{i}$ | |

2: Solve the SDP above. This will yield a POVM M. | |

3: Calculate the conditional states ${\gamma}_{i}^{S}$ for the POVM M and the optimal unitaries ${U}_{i}$. | |

4: repeat | ▹ Iterate steps 2 and 3 |

5: until convergence. |

## 5. The Role of Quantum Correlations

**Theorem**

**5.**

## 6. Multipartite Daemonic Ergotropy

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. POVM Advantage in Qutrit-Qubit Example

## Appendix B. Non-Optimal Convergence of the See-Saw Algorithm

## Appendix C. Proof of Theorem 5

**Theorem**

**A1.**

**Proof.**

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**Figure 1.**Illustration of daemonic ergotropy. A system S is coupled to an ancilla A. A measurement is performed on the latter and depending on the outcome i different unitaries can be applied to S in order to extract work. The maximal amount of extractable work using this protocol is the daemonic ergotropy.

**Figure 2.**Daemonic gain $\delta W$ as a function of the value of the highest energy level of the Hamiltonian H (in units of ${\u03f5}_{1})$ for the state ${\varrho}^{SA}$ given in Equation (5). Here $\u03f5={\u03f5}_{2}/{\u03f5}_{1}$. We compare the performance under the optimal r projective measurements (PVM) and positive operator valued measures (POVM). The latter was found numerically using the see-saw algorithm proposed here. The former is determined analytically as discussed in Appendix A. The dashed line is obtained as the daemonic gain $\delta W$ for the fixed POVM with effects ${E}_{j}=\frac{2}{3}\mathsf{\Pi}(2\pi j/3,0)$.

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**MDPI and ACS Style**

Bernards, F.; Kleinmann, M.; Gühne, O.; Paternostro, M.
Daemonic Ergotropy: Generalised Measurements and Multipartite Settings. *Entropy* **2019**, *21*, 771.
https://doi.org/10.3390/e21080771

**AMA Style**

Bernards F, Kleinmann M, Gühne O, Paternostro M.
Daemonic Ergotropy: Generalised Measurements and Multipartite Settings. *Entropy*. 2019; 21(8):771.
https://doi.org/10.3390/e21080771

**Chicago/Turabian Style**

Bernards, Fabian, Matthias Kleinmann, Otfried Gühne, and Mauro Paternostro.
2019. "Daemonic Ergotropy: Generalised Measurements and Multipartite Settings" *Entropy* 21, no. 8: 771.
https://doi.org/10.3390/e21080771