# Quantum Work Statistics with Initial Coherence

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Quantum Work Statistics

#### 2.2. Coherence Theory

## 3. Main Results

#### 3.1. Distance between the Averages of Work

**Theorem**

**1.**

#### 3.2. Distance between the Variances of Work

**Theorem**

**2.**

**Corollary**

**1.**

#### 3.3. Study of the Entropy Production

**Theorem**

**3.**

**Corollary**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

**Equation (8)**: The parameterisation of qudit states and unitaries makes finding an exact expression for the absolute difference between average works a difficult task to tackle. However, one can still find an upper bound to such difference as follows:$$\begin{array}{cc}\hfill |{\langle {w}_{\tau}\rangle}_{\mathrm{MH}}-{\langle {w}_{\tau}\rangle}_{\mathrm{TPM}}|& =\left(\right)open="|"\; close="|">\sum _{i\ne j}{\rho}_{ij}\langle j|{U}_{\tau}^{\u2020}\sum _{k}{h}_{k}|k\rangle \phantom{\rule{-0.166667em}{0ex}}\langle k|{U}_{\tau}|i\rangle \hfill \end{array}$$**Equation (9)**: When restricting our attention to qubits, we can parameterise unitary operations as ${U}_{\tau}={e}^{i\frac{\phi}{2}}\left(\begin{array}{cc}{e}^{i{\phi}_{1}}cos\tau & {e}^{i{\phi}_{2}}sin\tau \\ -{e}^{-i{\phi}_{2}}sin\tau & {e}^{-i{\phi}_{1}}cos\tau \end{array}\right)$, which generalises Equation (11). This gives us$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& |{\langle {w}_{\tau}\rangle}_{\mathrm{MH}}-{\langle {w}_{\tau}\rangle}_{\mathrm{TPM}}|=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}=\left(\right)open="|"\; close="|">\sum _{i\ne j}{\rho}_{ij}\langle j|{U}_{\tau}^{\u2020}\sum _{k}{h}_{k}|k\rangle \phantom{\rule{-0.166667em}{0ex}}\langle k|{U}_{\tau}|i\rangle =\left(\right)open="|"\; close="|">\sum _{i\ne j}{\rho}_{ij}\sum _{k=0,1}{h}_{k}{\gamma}_{ji}^{\left(k\right)}\hfill & =\left(\right)open="|"\; close="|">\sum _{i\ne j}{\rho}_{ij}\left(\right)open="("\; close=")">{h}_{0}{\gamma}_{ji}^{\left(0\right)}-{h}_{1}{\gamma}_{ji}^{\left(0\right)}\end{array}$$

## Appendix B. Proof of Theorem 2

**Equation (10)**: By using the triangle inequality, the definition of ${\gamma}^{\left(k\right)}$ given above and the fact that ${\gamma}_{ji}^{\left(k\right)}\le 1/2$, we can also provide the following upper bound$$\begin{array}{cc}\hfill |{\langle {w}_{\tau}^{2}\rangle}_{\mathrm{MH}}-{\langle {w}_{\tau}^{2}\rangle}_{\mathrm{TPM}}|& \le \sum _{i\ne j}|{\rho}_{ij}|\sum _{k}|{h}_{k}^{2}\left|\right|{\gamma}_{ji}^{\left(k\right)}|+\sum _{i\ne j}|{\rho}_{ij}|\sum _{l}|{h}_{l}\left|\right|{h}_{i}\left|\right|{\gamma}_{ji}^{\left(l\right)}|+\sum _{i\ne j}|{\rho}_{ij}|\sum _{m}|{h}_{j}\left|\right|{h}_{m}\left|\right|{\gamma}_{ji}^{\left(m\right)}|\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \frac{1}{2}{C}_{{l}_{1}}\left({\rho}_{0}\right)\mathrm{Tr}{H}^{2}+\frac{1}{2}{C}_{{l}_{1}}\left({\rho}_{0}\right)\underset{k}{max}|{h}_{k}\left|\mathrm{Tr}\right|H|+\frac{1}{2}{C}_{{l}_{1}}\left({\rho}_{0}\right)\underset{k}{max}|{h}_{k}\left|\mathrm{Tr}\right|H|.\hfill \end{array}$$- When focusing on qubits, we have$$\begin{array}{cc}\hfill |{\langle {w}_{\tau}^{2}\rangle}_{\mathrm{MH}}-{\langle {w}_{\tau}^{2}\rangle}_{\mathrm{TPM}}|& =\left(\right)open="|"\; close="|">\sum _{i\ne j}{\rho}_{ij}\langle j|{U}_{\tau}^{\u2020}{H}^{2}{U}_{\tau}-{U}_{\tau}^{\u2020}H{U}_{\tau}H-H{U}_{\tau}^{\u2020}H{U}_{\tau}|i\rangle \hfill \end{array}$$

## Appendix C. Derivation of Table 1

## Appendix D. Proof of Theorem 3

**Equation (17)**: The Jarzynski equality [42] ${\langle {e}^{-\beta ({w}_{\tau}-\Delta {F}_{\tau})}\rangle}_{\mathrm{TPM}}=1$ is only fulfilled when the initial state is at equilibrium. For an arbitrary initial state $\rho $, the following fluctuation Theorem applies [21]$${\langle {e}^{-\beta ({w}_{\tau}-\Delta {F}_{\tau})}\rangle}_{\mathrm{MH}}=\mathrm{Re}\left(\right)open="("\; close=")">\mathrm{Tr}\left[{\gamma}_{\tau}{\mathcal{G}}_{0}^{-1}{\rho}_{0}\right]$$$$\Delta {F}_{\tau}=-(ln{\langle {e}^{-\beta {w}_{\tau}}\rangle}_{\mathrm{MH}}-ln{\xi}_{\tau})/\beta .$$We use this result in the definition of entropy production $\mathsf{\Sigma}=\beta (w-\Delta F)$ and use a cumulant expansion of ${\langle {e}^{-\beta {w}_{\tau}}\rangle}_{\mathrm{MH}}$ to find [44]$${\langle {\mathsf{\Sigma}}_{\tau}\rangle}_{\mathrm{MH}}=\sum _{n\ge 2}\frac{{(-1)}^{n}}{n!}{\kappa}_{\tau}^{\left(n\right)}\left(\beta \right){\beta}^{n}-ln{\xi}_{\tau},$$In the linear response regime, the first term yields $\frac{{\beta}^{2}}{2}{(\Delta {w}_{\tau})}_{\mathrm{MH}}^{2}$ [44], where ${(\Delta {w}_{\tau})}_{\mathrm{MH}}^{2}={\kappa}_{\tau}^{\left(2\right)}\left(\beta \right)$ is the variance of the MH distribution of work. Expanding the second term gives$$\begin{array}{cc}\hfill ln{\xi}_{\tau}\phantom{\rule{1.em}{0ex}}& \approx \frac{{\beta}^{2}}{4}\mathrm{Tr}[{H}_{0}^{2}-{H}_{\tau}^{2}]-\beta {\langle {w}_{\tau}\rangle}_{\mathrm{MH}}+\frac{{\beta}^{2}}{2}\left(\right)open="\{"\; close="\}">{(\Delta {w}_{\tau})}_{\mathrm{MH}}^{2}+\mathrm{Re}\left(\right)open="("\; close=")">\mathrm{Tr}\left[{\rho}_{0}[{H}_{0},{U}_{\tau}^{\u2020}{H}_{\tau}{U}_{\tau}]\right]\hfill & .\end{array}$$Thus, the average entropy production in the linear response regime amounts to$${\langle {\mathsf{\Sigma}}_{\tau}\rangle}_{\mathrm{MH}}^{\mathrm{LR}}=\beta {\langle {w}_{\tau}\rangle}_{\mathrm{MH}}-{\displaystyle \frac{{\beta}^{2}}{2}}\mathrm{Re}\left(\right)open="("\; close=")">\mathrm{Tr}\left[{\rho}_{0}[{H}_{0},{U}_{\tau}^{\u2020}{H}_{\tau}{U}_{\tau}]\right]$$**Equation (18)**: Let us now have a close look at qubits. For convenience, we consider a qubit prepared in the state $\rho =\frac{1}{2}\left(\begin{array}{cc}1-{a}_{z}& {a}_{x}-i{a}_{y}\\ {a}_{x}+i{a}_{y}& 1+{a}_{z}\end{array}\right)$, with $\mathit{a}\in {\mathbb{R}}^{3}$, $\frac{1-{a}_{z}}{2}}={\displaystyle \frac{{e}^{-\beta}}{\mathrm{Tr}\left[{e}^{-\beta {H}_{0}}\right]}$ and ${H}_{0}={\sigma}_{z}$. This ensures that, for small enough $\beta $, ${a}_{z}$ will also be small:$$\frac{1-{a}_{z}}{2}}\approx {\displaystyle \frac{1-\beta}{2}}\to {a}_{z}\approx \beta .$$Let us suppose the qubit is subjected to a real unitary transformation ${U}_{\tau}$ such that ${H}_{\tau}=k{\sigma}_{z}$, for $k\in \mathbb{C}$. The average entropy production in the linear response regime is then given by Equation (17)$$\begin{array}{cc}\hfill {\langle {\mathsf{\Sigma}}_{\tau}\rangle}_{\mathrm{MH}}^{\mathrm{LR}}& ={a}_{x}\beta ksin\left(2\tau \right)-2{a}_{z}\beta kcos{\left(\tau \right)}^{2}+\frac{{\beta}^{2}{k}^{2}}{2}+{a}_{z}\beta k-\frac{{\beta}^{2}}{2}+{a}_{z}\beta \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \approx \beta ksin\left(2\tau \right)cos\left(\chi \right){C}_{{l}_{1}}\left({\rho}_{0}\right)-2{\beta}^{2}kcos{\left(\tau \right)}^{2}+\frac{{\beta}^{2}{k}^{2}}{2}+{\beta}^{2}k-\frac{{\beta}^{2}}{2}+{\beta}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\beta ksin\left(2\tau \right)cos\left(\chi \right){C}_{{l}_{1}}\left({\rho}_{0}\right)-2{\beta}^{2}kcos{\left(\tau \right)}^{2}+\frac{{\beta}^{2}{k}^{2}}{2}+{\beta}^{2}k+\frac{{\beta}^{2}}{2},\hfill \end{array}$$$$\begin{array}{cc}\hfill {\langle {\mathsf{\Sigma}}_{\tau}\rangle}_{\mathrm{TPM}}^{\mathrm{LR}}& =2{\beta}^{2}k(1-cos{\left(\tau \right)}^{2})-2{a}_{z}^{2}{\beta}^{2}{k}^{2}sin{\left(\tau \right)}^{4}+2{a}_{z}^{2}{\beta}^{2}{k}^{2}sin{\left(\tau \right)}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -2{a}_{z}^{2}{\beta}^{2}ksin{\left(\tau \right)}^{2}-\frac{{a}_{z}^{2}{\beta}^{2}{k}^{2}}{2}+\frac{{\beta}^{2}{k}^{2}}{2}+{a}_{z}^{2}{\beta}^{2}k-{\beta}^{2}k-\frac{{a}_{z}^{2}{\beta}^{2}}{2}+\frac{{\beta}^{2}}{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \approx -2{\beta}^{2}kcos{\left(\tau \right)}^{2}+\frac{{\beta}^{2}{k}^{2}}{2}+{\beta}^{2}k+\frac{{\beta}^{2}}{2},\hfill \end{array}$$$${\langle {\mathsf{\Sigma}}_{\tau}\rangle}_{\mathrm{MH}}^{\mathrm{LR}}=\beta ksin\left(2\tau \right)cos\left(\chi \right){C}_{{l}_{1}}\left({\rho}_{0}\right)+{\langle {\mathsf{\Sigma}}_{\tau}\rangle}_{\mathrm{TPM}}^{\mathrm{LR}}.$$

**Figure A1.**Qubit in the initial state where ${\alpha}^{2}={\displaystyle \frac{{e}^{\beta}}{\mathrm{Tr}{e}^{-\beta H\left({\lambda}_{0}\right)}}}$, $0\le \omega \le 1$ and $H\left({\lambda}_{0}\right)={\sigma}_{z}$, undergoing an evolution given by a real unitary ${U}_{\tau}=\left(\begin{array}{cc}cos\left(\tau \right)& sin\left(\tau \right)\\ -sin\left(\tau \right)& cos\left(\tau \right)\end{array}\right)$ and $H\left({\lambda}_{\tau}\right)=\frac{1}{2}{\sigma}_{z}$. (

**a**) $\langle {\mathsf{\Sigma}}_{\tau}\rangle $ versus initial coherence, $\beta =0.2$, $\tau =\frac{3\pi}{4}$. MH scheme (green) and TPM scheme (red). (

**b**) ${\langle {w}_{\tau}\rangle}_{\mathrm{MH}}$ (green), ${\langle {w}_{\tau}\rangle}_{\mathrm{TPM}}$ (blue line), $\Delta {F}_{\tau ,\mathrm{MH}}$ (red), and $\Delta {F}_{\tau ,\mathrm{TPM}}$ (blue circles) (naturally, they both agree), versus initial coherence, $\beta =0.2$, $\tau =\frac{3\pi}{4}$. (

**c**) $\langle {\mathsf{\Sigma}}_{\tau}\rangle $ versus initial coherence, $\beta =0.2$, $\tau =\frac{3\pi}{4}$. MH scheme (green) and TPM scheme (red). Negativity of the MH distribution, computed as ${min}_{mn}\mathrm{Re}\mathrm{Tr}\left({U}_{\tau}^{\u2020}\right|n\rangle \phantom{\rule{-0.166667em}{0ex}}\langle n|{U}_{\tau}|m\rangle \phantom{\rule{-0.166667em}{0ex}}\langle m\left|{\rho}_{0}\right)$ (blue). (

**d**) $\langle {\mathsf{\Sigma}}_{\tau}\rangle $ versus initial coherence, $\beta =0.2$, $\tau =\frac{3\pi}{4}$. MH scheme (green) and TPM scheme (red). ${\langle {e}^{-\beta {w}_{\tau}}\rangle}_{\mathrm{MH}}$ (magenta) and ${\xi}_{\tau}$ (blue).

## Appendix E. Proof of Corollary 2

## Appendix F. Why Can 〈Σ_{τ}〉_{MH} Be Negative?

## References

- Thermodynamics in the Quantum Regime; Binder, F.; Correa, L.; Gogolin, C.; Anders, J.; Adesso, G. (Eds.) Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar]
- Deffner, S.; Campbell, S. Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information; Morgan & Claypool Publishers: San Rafael, CA, USA, 2019. [Google Scholar]
- Santos, J.P.; Céleri, L.C.; Landi, G.T.; Paternostro, M. The role of quantum coherence in non-equilibrium entropy production. NPJ Quant. Inf.
**2019**, 5, 23. [Google Scholar] [CrossRef][Green Version] - Francica, G.; Goold, J.; Plastina. The role of coherence in the non-equilibrium thermodynamics of quantum systems. Phys. Rev. E
**2019**, 99, 042105. [Google Scholar] [CrossRef][Green Version] - Riechers, P.M.; Gu, M. Initial-State Dependence of Thermodynamic Dissipation for any Quantum Process. arXiv
**2020**, arXiv:2002.11425. [Google Scholar] - Sone, A.; Liu, Y.X.; Cappellaro, P. Quantum Jarzynski equality in open quantum systems from the one-time measurement scheme. arXiv
**2020**, arXiv:2002.06332. [Google Scholar] - Francica, G.; Binder, F.; Guarnieri, G.; Mitchison, M.T.; Goold, J.; Plastina, F. Quantum coherence and ergotropy. arXiv
**2020**, arXiv:2006.05424. [Google Scholar] - Francica, G.; Goold, J.; Plastina, F.; Paternostro, M. Daemonic Ergotropy: Enhanced Work Extraction from Quantum Correlations. NPJ Quant. Inf.
**2017**, 3, 12. [Google Scholar] [CrossRef][Green Version] - Bernards, F.; Kleinmann, M.; Gühne, O.; Paternostro, M. Daemonic Ergotropy: Generalised Measurements and Multipartite Settings. Entropy
**2019**, 21, 771. [Google Scholar] [CrossRef][Green Version] - Miller, H.J.D.; Mohammady, M.H.; Perarnau-Llobet, M.; Guarnieri, G. Thermodynamic uncertainty relation in slowly driven quantum heat engines. arXiv
**2020**, arXiv:2006.07316. [Google Scholar] - Miller, H.J.; Scandi, M.; Anders, J.; Perarnau-Llobet, M. Work Fluctuations in Slow Processes: Quantum Signatures and Optimal Control. Phys. Rev. Lett.
**2019**, 123. [Google Scholar] [CrossRef][Green Version] - Scandi, M.; Miller, H.J.D.; Anders, J.; Perarnau-Llobet, M. Quantum work statistics close to equilibrium. arXiv
**2019**, arXiv:1911.04306. [Google Scholar] [CrossRef] - Talkner, P.; Lutz, E.; Hänggi, P. Fluctuation theorems: Work is not an observable. Phys. Rev. E
**2007**, 75, 050102. [Google Scholar] [CrossRef][Green Version] - Campisi, M.; Hänggi, P.; Talkner, P. Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys.
**2011**, 83, 771. [Google Scholar] [CrossRef][Green Version] - Esposito, M.; Harbola, U.; Mukamel, S. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum system. Rev. Mod. Phys.
**2009**, 81, 1665. [Google Scholar] [CrossRef][Green Version] - Batalhão, T.B.; Souza, A.M.; Mazzola, L.; Auccaise, R.; Sarthour, R.S.; Oliveira, I.S.; Goold, J.; De Chiara, G.; Paternostro, M.; Serra, R.M. Experimental reconstruction of work distribution and study of fluctuation relations in a closed quantum system. Phys. Rev. Lett.
**2014**, 113, 140601. [Google Scholar] [CrossRef][Green Version] - An, S.; Zhang, J.N.; Um, M.; Lv, D.; Lu, Y.; Zhang, J.; Yin, Z.Q.; Quan, H.; Kim, K. Experimental test of the quantum Jarzynski equality with a trapped-ion system. Nat. Phys.
**2015**, 11, 193–199. [Google Scholar] [CrossRef][Green Version] - Peterson, J.P.; Batalhão, T.B.; Herrera, M.; Souza, A.M.; Sarthour, R.S.; Oliveira, I.S.; Serra, R.M. Experimental characterization of a spin quantum heat engine. Phys. Rev. Lett.
**2019**, 123, 240601. [Google Scholar] [CrossRef][Green Version] - Ronzani, A.; Karimi, B.; Senior, J.; Chang, Y.C.; Peltonen, J.T.; Chen, C.; Pekola, J.P. Tunable photonic heat transport in a quantum heat valve. Nat. Phys.
**2018**, 14, 991–995. [Google Scholar] [CrossRef][Green Version] - von Lindenfels, D.; Gräb, O.; Schmiegelow, C.T.; Kaushal, V.; Schulz, J.; Mitchison, M.T.; Goold, J.; Schmidt-Kaler, F.; Poschinger, U.G. Spin Heat Engine Coupled to a Harmonic-Oscillator Flywheel. Phys. Rev. Lett.
**2019**, 123, 080602. [Google Scholar] [CrossRef][Green Version] - Allahverdyan, A.E. Nonequilibrium quantum fluctuations of work. Phys. Rev. E
**2014**, 90, 032137. [Google Scholar] [CrossRef][Green Version] - Micadei, K.; Landi, G.T.; Lutz, E. Quantum Fluctuation Theorems beyond Two-Point Measurements. Phys. Rev. Lett.
**2020**, 124, 090602. [Google Scholar] [CrossRef][Green Version] - Levy, A.; Lostaglio, M. A quasiprobability distribution for heat fluctuations in the quantum regime. arXiv
**2019**, arXiv:1909.11116. [Google Scholar] [CrossRef] - Gherardini, S.; Belenchia, A.; Paternostro, M.; Trombettoni, A. The role of quantum coherence in energy fluctuations. arXiv
**2020**, arXiv:2006.06208. [Google Scholar] - Solinas, P.; Gasparinetti, S. Full distribution of work done on a quantum system for arbitrary initial states. Phys. Rev. E
**2015**, 92, 042150. [Google Scholar] - de Falco, D.; Tamascelli, D. Noise-assisted quantum transport and computation. J. Phys. A Math. Theor.
**2013**, 22, 5301–5316. [Google Scholar] [CrossRef][Green Version] - Solinas, P.; Gasparinetti, S. Probing quantum interference effects in the work distribution. Phys. Rev. A
**2016**, 94, 052103. [Google Scholar] [CrossRef][Green Version] - Terletsky, Y.P. The limiting transition from quantum to classical mechanics. J. Exp. Theor. Phys.
**1937**, 7, 1290. [Google Scholar] - Margenau, H.; Hill, R.N. Correlation between measurements in quantum theory. Prog. Theor. Phys.
**1961**, 26, 722. [Google Scholar] [CrossRef][Green Version] - Kirkwood, J.G. Quantum statistics of almost classical assemblies. Phys. Rev.
**1933**, 44, 37. [Google Scholar] [CrossRef] - Baumgratz, T.; Cramer, M.; Plenio, M.B. Quantifying Coherence. Phys. Rev. Lett.
**2014**, 113, 140401. [Google Scholar] [CrossRef][Green Version] - Miller, H.J.D.; Anders, J. Time-reversal symmetric work distributions for closed quantum dynamics in the histories framework. New J. Phys.
**2017**, 19, 062001. [Google Scholar] [CrossRef] - Åberg, J. Quantifying Superposition. arXiv
**2006**, arXiv:0612146. [Google Scholar] - Braun, D.; Georgeot, B. Quantitative measure of interference. Phys. Rev. A
**2006**, 73, 022314. [Google Scholar] [CrossRef][Green Version] - Streltsov, A.; Adesso, G.; Plenio, M.B. Colloquium: Quantum coherence as a resource. Rev. Mod. Phys.
**2017**, 89, 041003. [Google Scholar] [CrossRef][Green Version] - Winter, A.; Yang, D. Operational Resource Theory of Coherence. Phys. Rev. Lett.
**2016**, 116, 120404. [Google Scholar] [CrossRef][Green Version] - Brandão, F.G.S.L.; Gour, G. Reversible Framework for Quantum Resource Theories. Phys. Rev. Lett.
**2015**, 115, 070503. [Google Scholar] [CrossRef][Green Version] - Bu, K.; Xiong, C. A Note on Cohering Power and De-cohering Power. Quant. Inf. Comput.
**2017**, 17, 1206–1220. [Google Scholar] - Ding, X.; Yi, J.; Kim, Y.W.; Talkner, P. Measurement-driven single temperature engine. Phys. Rev. E
**2018**, 98, 042122. [Google Scholar] [CrossRef][Green Version] - Elouard, C.; Herrera-Martí, D.; Huard, B.; Auffèves, A. Extracting Work from Quantum Measurement in Maxwell’s Demon Engines. Phys. Rev. Lett.
**2017**, 118, 260603. [Google Scholar] [CrossRef] - Buffoni, L.; Solfanelli, A.; Verrucchi, P.; Cuccoli, A.; Campisi, M. Quantum Measurement Cooling. Phys. Rev. Lett.
**2019**, 122. [Google Scholar] [CrossRef][Green Version] - Jarzynski, C. Nonequilibrium equality for free energy differences. Phys. Rev. Lett.
**1997**, 78, 2690–2693. [Google Scholar] [CrossRef][Green Version] - Kubo, R. Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn.
**1957**, 12, 570–586. [Google Scholar] [CrossRef] - Batalhão, T.B.; Souza, A.M.; Sarthour, R.S.; Oliveira, I.S.; Paternostro, M.; Lutz, E.; Serra, R.M. Irreversibility and the arrow of time in a quenched quantum system. Phys. Rev. Lett.
**2015**, 115, 190601. [Google Scholar] [CrossRef] [PubMed][Green Version]

**Figure 1.**Equatorial plane of the Bloch sphere at $z=0$. The ${l}_{1}$-coherence of a state quantifies its distance from the z-axis.

**Figure 2.**(

**a**) Maximum absolute distance between the first moments of work obtained via the Margenau–Hill (MH) scheme and the two-point measurement (TPM) scheme (red dots) and the bound in Equation (8) (blue crosses) versus initial coherence. (

**b**) Maximum absolute distance between the second moments of work obtained via the MH scheme and the TPM scheme (red dots) and bound (10) (blue crosses) versus initial coherence. Each point represents a simulation for a different random initial state ${\rho}_{0}$. Both panels refer to a $d=3$ system governed by the Hamiltonian $H=(1/\sqrt{3})\mathrm{Diag}[1,1,-2]$.

**Figure 3.**We plot the discrepancy ${(\Delta w)}_{\mathrm{MH}}^{2}-{(\Delta w)}_{\mathrm{TPM}}^{2}$ between the variances of the TPM and MH distributions for pure states of $d=2$ systems with Bloch vector $({a}_{x},0,\sqrt{1-{a}_{x}^{2}})$ against ${a}_{x}$. We have taken ${H}_{0}={H}_{\tau}={\sigma}_{z}$ and the dynamics described by Equation (11). We have taken $\tau =0.1$ (red), $\tau =\pi /5$ (magenta), $\tau =\pi /4$ (green), and $\tau =3\pi /4$ (blue).

**Figure 4.**We correlate the value of the discrepancy ${(\Delta {w}_{\tau})}_{\mathrm{MH}}^{2}-{(\Delta {w}_{\tau})}_{\mathrm{TPM}}^{2}$ between the variances of the TPM and MH distributions to the specific point on the Bloch sphere that represents a pure state of $d=2$ systems. We have used ${H}_{0}={H}_{\tau}={\sigma}_{z}$, the unitary propagator in Equation (11), and $\tau =0.1$ (panel (

**a**)), $\tau =\pi /4$ (panel (

**b**)), $\tau =\pi /2-0.01$ (panel (

**c**)), and $\tau =3\pi /4$ (panel (

**d**)).

**Figure 5.**(

**a**) We plot the average entropy production $\langle {\mathsf{\Sigma}}_{\tau}\rangle $ versus time for $\beta =0.2$, $\omega =0.8$. The behavior corresponding to the MH (TPM) scheme is shown by the green (red) curve. (

**b**) We show ${min}_{\omega}\langle {\mathsf{\Sigma}}_{\tau}\rangle $ for the MH and TPM schemes (green and red curves, respectively), and $-log\xi $ (blue curve) versus $\beta $ for $\tau =3\pi /4$.

**Table 1.**Relation between the variances of the MH and the TPM schemes for a two-level system with a Bloch vector with ${a}_{y}=0$ and dynamics ruled by Equation (11). In Table (

**a**), we have taken $\mathrm{sgn}\left({a}_{z}\right)=\mathrm{sgn}(tan\tau )$, while Table (

**b**) is for $\mathrm{sgn}\left({a}_{z}\right)\ne \mathrm{sgn}(tan\tau )$.

$\mathbf{sgn}\left({\mathit{a}}_{\mathit{z}}\right)=\mathbf{sgn}(tan\mathit{\tau})$ | ||
---|---|---|

${a}_{x}\in [-1,-2{a}_{z}tan\tau ]$ | ${a}_{x}\in [-2{a}_{z}tan\tau ,0]$ | ${a}_{x}\in [0,1]$ |

${(\Delta w)}_{\mathrm{MH}}^{2}\le {(\Delta w)}_{\mathrm{TPM}}^{2}$ | ${(\Delta w)}_{\mathrm{MH}}^{2}\ge {(\Delta w)}_{\mathrm{TPM}}^{2}$ | ${(\Delta w)}_{\mathrm{MH}}^{2}\le {(\Delta w)}_{\mathrm{TPM}}^{2}$ |

(a) | ||

$\mathrm{sgn}\left({a}_{z}\right)\ne \mathrm{sgn}(tan\tau )$ | ||

${a}_{x}\in [-1,0]$ | ${a}_{x}\in [0,-2{a}_{z}tan\tau ]$ | ${a}_{x}\in [-2{a}_{z}tan\tau ,1]$ |

${(\Delta w)}_{\mathrm{MH}}^{2}\le {(\Delta w)}_{\mathrm{TPM}}^{2}$ | ${(\Delta w)}_{\mathrm{MH}}^{2}\ge {(\Delta w)}_{\mathrm{TPM}}^{2}$ | ${(\Delta w)}_{\mathrm{MH}}^{2}\le {(\Delta w)}_{\mathrm{TPM}}^{2}$ |

(b) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Díaz, M.G.; Guarnieri, G.; Paternostro, M.
Quantum Work Statistics with Initial Coherence. *Entropy* **2020**, *22*, 1223.
https://doi.org/10.3390/e22111223

**AMA Style**

Díaz MG, Guarnieri G, Paternostro M.
Quantum Work Statistics with Initial Coherence. *Entropy*. 2020; 22(11):1223.
https://doi.org/10.3390/e22111223

**Chicago/Turabian Style**

Díaz, María García, Giacomo Guarnieri, and Mauro Paternostro.
2020. "Quantum Work Statistics with Initial Coherence" *Entropy* 22, no. 11: 1223.
https://doi.org/10.3390/e22111223