# Quantifying Athermality and Quantum Induced Deviations from Classical Fluctuation Relations

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Classical Fluctuation Relations

#### 2.2. Fully Quantum Fluctuation Relations

## 3. Results

#### 3.1. Photon Added and Subtracted Thermal States

#### 3.2. Binomial States

- Forwards: The battery B is prepared in the state $|n,{\tilde{p}}_{i}\rangle $ and measured in $|n,{p}_{f}\rangle $
- Reverse: The battery B is prepared in the state $|n,{\tilde{p}}_{f}\rangle $ and measured in $|n,{p}_{i}\rangle $.

- Forwards: The battery B is prepared in the state $|{n}_{i},\tilde{p}\rangle $ and measured in $|{n}_{f},p\rangle $.
- Reverse: The battery B is prepared in the state $|{n}_{f},\tilde{p}\rangle $ and measured in $|{n}_{i},p\rangle $.

#### 3.3. Energy Translation Invariance, Jarzynski Relations and Stochastic Entropy Production

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of Photon Added and Subtracted Crooks Equality

## Appendix B. Derivation of Binomial State Properties

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### Appendix B.1. The Quantum Distortion Factor for Binomial States

#### Appendix B.2. The Harmonic Limit

## References

- Guggenheim, E.A. The Thermodynamics of Magnetization. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1936**, 155, 70–101. [Google Scholar] [CrossRef][Green Version] - Alloul, H. Thermodynamics of Superconductors; Springer: Berlin/Heidelberg, Germany, 2011; pp. 175–199. [Google Scholar] [CrossRef]
- Page, D.N. Hawking radiation and black hole thermodynamics. New J. Phys.
**2005**, 7, 203. [Google Scholar] [CrossRef] - Schrödinger, E.; Penrose, R. What is Life?: With Mind and Matter and Autobiographical Sketches; Canto, Cambridge University Press: Cambridge, UK, 1992. [Google Scholar] [CrossRef]
- Haynie, D. Biological Thermodynamics; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Ott, J.; Boerio-Goates, J. Chemical Thermodynamics: Principles and Applications: Principles and Applications; Elsevier Science: Amsterdam, The Netherlands, 2000. [Google Scholar]
- Callen, H.; Callen, H.; Sons, W. Thermodynamics and an Introduction to Thermostatistics; Wiley: Hoboken, NJ, USA, 1985. [Google Scholar]
- Jarzynski, C. Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale. Annu. Rev. Condens. Matter Phys.
**2011**, 2, 329–351. [Google Scholar] [CrossRef][Green Version] - Crooks, G.E. Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E
**1999**, 60, 2721–2726. [Google Scholar] [CrossRef][Green Version] - Talkner, P.; Hänggi, P. The Tasaki–Crooks quantum fluctuation theorem. J. Phys. Math. Theor.
**2007**, 40, F569. [Google Scholar] [CrossRef] - Evans, D.J.; Searles, D.J. Equilibrium microstates which generate second law violating steady states. Phys. Rev. E
**1994**, 50, 1645–1648. [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] - Tasaki, H. Jarzynski Relations for Quantum Systems and Some Applications. arXiv
**2000**, arXiv:cond-mat/0009244. [Google Scholar] - Kurchan, J. A Quantum Fluctuation Theorem. arXiv
**2000**, arXiv:cond-mat/0007360. [Google Scholar] - Hänggi, P.; Talkner, P. The other QFT. Nat. Phys.
**2015**, 11, 108. [Google Scholar] [CrossRef] - Esposito, M.; Harbola, U.; Mukamel, S. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys.
**2009**, 81, 1665–1702. [Google Scholar] [CrossRef][Green Version] - Campisi, M.; Hänggi, P.; Talkner, P. Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys.
**2011**, 83, 771–791. [Google Scholar] [CrossRef][Green Version] - Albash, T.; Lidar, D.A.; Marvian, M.; Zanardi, P. Fluctuation theorems for quantum processes. Phys. Rev. E
**2013**, 88, 032146. [Google Scholar] [CrossRef][Green Version] - Manzano, G.; Horowitz, J.M.; Parrondo, J.M.R. Nonequilibrium potential and fluctuation theorems for quantum maps. Phys. Rev. E
**2015**, 92, 032129. [Google Scholar] [CrossRef][Green Version] - Rastegin, A.E. Non-equilibrium equalities with unital quantum channels. J. Stat. Mech. Theory Exp.
**2013**, 2013, P06016. [Google Scholar] [CrossRef] - Campisi, M.; Talkner, P.; Hänggi, P. Fluctuation Theorem for Arbitrary Open Quantum Systems. Phys. Rev. Lett.
**2009**, 102, 210401. [Google Scholar] [CrossRef] - Jarzynski, C. Nonequilibrium work theorem for a system strongly coupled to a thermal environment. J. Stat. Mech. Theory Exp.
**2004**, 2004, P09005. [Google Scholar] [CrossRef] - Solinas, P.; Miller, H.J.D.; Anders, J. Measurement-dependent corrections to work distributions arising from quantum coherences. Phys. Rev. A
**2017**, 96, 052115. [Google Scholar] [CrossRef][Green Version] - Allahverdyan, A.E. Nonequilibrium quantum fluctuations of work. Phys. Rev. E
**2014**, 90, 032137. [Google Scholar] [CrossRef] [PubMed][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] - Elouard, C.; Mohammady, M.H. Work, Heat and Entropy Production Along Quantum Trajectories. In Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions; Binder, F., Correa, L.A., Gogolin, C., Anders, J., Adesso, G., Eds.; Springer International Publishing: Cham, Switzerland, 2018; pp. 363–393. [Google Scholar] [CrossRef][Green Version]
- Horowitz, J.M. Quantum-trajectory approach to the stochastic thermodynamics of a forced harmonic oscillator. Phys. Rev. E
**2012**, 85, 31110. [Google Scholar] [CrossRef] [PubMed][Green Version] - Elouard, C.; Herrera-Martí, D.A.; Clusel, M.; Auffèves, A. The role of quantum measurement in stochastic thermodynamics. npj Quantum Inf.
**2017**, 3, 9. [Google Scholar] [CrossRef] - Åberg, J. Fully Quantum Fluctuation Theorems. Phys. Rev. X
**2018**, 8, 11019. [Google Scholar] [CrossRef][Green Version] - Alhambra, A.M.; Masanes, L.; Oppenheim, J.; Perry, C. Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality. Phys. Rev. X
**2016**, 6, 041017. [Google Scholar] [CrossRef][Green Version] - Kwon, H.; Kim, M.S. Fluctuation Theorems for a Quantum Channel. Phys. Rev. X
**2019**, 9, 031029. [Google Scholar] [CrossRef][Green Version] - Holmes, Z. The Coherent Crooks Equality. In Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions; Binder, F., Correa, L.A., Gogolin, C., Anders, J., Adesso, G., Eds.; Springer International Publishing: Cham, Switzerland, 2018; pp. 301–316. [Google Scholar] [CrossRef][Green Version]
- Horodecki, M.; Oppenheim, J. Fundamental limitations for quantum and nanoscale thermodynamics. Nat. Commun.
**2013**, 4, 2059. [Google Scholar] [CrossRef] - Brandão, F.; Horodecki, M.; Ng, N.; Oppenheim, J.; Wehner, S. The second laws of quantum thermodynamics. Proc. Natl. Acad. Sci. USA
**2015**, 112, 3275–3279. [Google Scholar] [CrossRef][Green Version] - Åberg, J. Truly work-like work extraction via a single-shot analysis. Nat. Commun.
**2013**, 4, 1925. [Google Scholar] [CrossRef][Green Version] - Gour, G.; Jennings, D.; Buscemi, F.; Duan, R.; Marvian, I. Quantum majorization and a complete set of entropic conditions for quantum thermodynamics. Nat. Commun.
**2018**, 9, 5352. [Google Scholar] [CrossRef] - Lostaglio, M.; Jennings, D.; Rudolph, T. Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nat. Commun.
**2015**, 6, 6383. [Google Scholar] [CrossRef][Green Version] - Åberg, J. Catalytic Coherence. Phys. Rev. Lett.
**2014**, 113, 150402. [Google Scholar] [CrossRef] [PubMed][Green Version] - Korzekwa, K.; Lostaglio, M.; Oppenheim, J.; Jennings, D. The extraction of work from quantum coherence. New J. Phys.
**2016**, 18, 023045. [Google Scholar] [CrossRef] - Holmes, Z.; Weidt, S.; Jennings, D.; Anders, J.; Mintert, F. Coherent fluctuation relations: From the abstract to the concrete. Quantum
**2019**, 3, 124. [Google Scholar] [CrossRef] - Mingo, E.H.; Jennings, D. Decomposable coherence and quantum fluctuation relations. Quantum
**2019**, 3, 202. [Google Scholar] [CrossRef] - Zavatta, A.; Parigi, V.; Kim, M.S.; Bellini, M. Subtracting photons from arbitrary light fields: Experimental test of coherent state invariance by single-photon annihilation. New J. Phys.
**2008**, 10, 123006. [Google Scholar] [CrossRef] - Ueda, M.; Imoto, N.; Ogawa, T. Quantum theory for continuous photodetection processes. Phys. Rev. A
**1990**, 41, 3891–3904. [Google Scholar] [CrossRef] - Barnett, S.M.; Ferenczi, G.; Gilson, C.R.; Speirits, F.C. Statistics of photon-subtracted and photon-added states. Phys. Rev. A
**2018**, 98, 013809. [Google Scholar] [CrossRef][Green Version] - Zavatta, A.; Parigi, V.; Bellini, M. Experimental nonclassicality of single-photon-added thermal light states. Phys. Rev. A
**2007**, 75, 052106. [Google Scholar] [CrossRef][Green Version] - Hlousek, J.; Jezek, M.; Filip, R. Work and information from thermal states after subtraction of energy quanta. Sci. Rep.
**2017**, 7, 13046. [Google Scholar] [CrossRef][Green Version] - Vidrighin, M.D.; Dahlsten, O.; Barbieri, M.; Kim, M.S.; Vedral, V.; Walmsley, I.A. Photonic Maxwell’s Demon. Phys. Rev. Lett.
**2016**, 116, 050401. [Google Scholar] [CrossRef][Green Version] - Quantum communication with photon-added coherent states. Quantum Inf. Process.
**2013**, 12, 537–547. [CrossRef] - Braun, D.; Jian, P.; Pinel, O.; Treps, N. Precision measurements with photon-subtracted or photon-added Gaussian states. Phys. Rev. A
**2014**, 90, 013821. [Google Scholar] [CrossRef][Green Version] - Mari, A.; Eisert, J. Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient. Phys. Rev. Lett.
**2012**, 109, 230503. [Google Scholar] [CrossRef][Green Version] - Walschaers, M.; Fabre, C.; Parigi, V.; Treps, N. Entanglement and Wigner Function Negativity of Multimode Non-Gaussian States. Phys. Rev. Lett.
**2017**, 119, 183601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Perelomov, A. Generalized Coherent States and Their Applications; Theoretical and Mathematical Physics; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Arecchi, F.T.; Courtens, E.; Gilmore, R.; Thomas, H. Atomic Coherent States in Quantum Optics. Phys. Rev. A
**1972**, 6, 2211–2237. [Google Scholar] [CrossRef] - Stoler, D.; Saleh, B.; Teich, M. Binomial States of the Quantized Radiation Field. Opt. Acta Int. J. Opt.
**1985**, 32, 345–355. [Google Scholar] [CrossRef] - Vidiella-Barranco, A.; Roversi, J.A. Statistical and phase properties of the binomial states of the electromagnetic field. Phys. Rev. A
**1994**, 50, 5233–5241. [Google Scholar] [CrossRef][Green Version] - Maleki, Y.; Maleki, A. Entangled multimode spin coherent states of trapped ions. J. Opt. Soc. Am. B
**2018**, 35, 1211–1217. [Google Scholar] [CrossRef] - Miry, S.R.; Tavassoly, M.K.; Roknizadeh, R. On the generation of number states, their single- and two-mode superpositions, and two-mode binomial state in a cavity. J. Opt. Soc. Am. B
**2014**, 31, 270–276. [Google Scholar] [CrossRef] - Garling, D.J.H. Inequalities: A Journey into Linear Analysis; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar] [CrossRef]
- Baştuğ, T.; Kuyucak, S. Application of Jarzynski’s equality in simple versus complex systems. Chem. Phys. Lett.
**2007**, 436, 383–387. [Google Scholar] [CrossRef] - West, D.K.; Olmsted, P.D.; Paci, E. Free energy for protein folding from nonequilibrium simulations using the Jarzynski equality. J. Chem. Phys.
**2006**, 125, 204910. [Google Scholar] [CrossRef] [PubMed] - Gittes, F. Two famous results of Einstein derived from the Jarzynski equality. Am. J. Phys.
**2018**, 86, 31–35. [Google Scholar] [CrossRef][Green Version] - Deffner, S.; Jarzynski, C. Information Processing and the Second Law of Thermodynamics: An Inclusive, Hamiltonian Approach. Phys. Rev. X
**2013**, 3, 041003. [Google Scholar] [CrossRef][Green Version] - Ballentine, L. Quantum Mechanics: A Modern Development; World Scientific: Singapore, 1998. [Google Scholar]
- Crooks, G.E. Quantum operation time reversal. Phys. Rev. A
**2008**, 77, 034101. [Google Scholar] [CrossRef][Green Version] - Jones, G.N.; Haight, J.; Lee, C.T. Nonclassical effects in the photon-added thermal state. Quantum Semiclassical Opt. J. Eur. Soc. Part B
**1997**, 9, 411–418. [Google Scholar] [CrossRef] - Usha Devi, A.R.; Prabhu, R.; Uma, M.S. Non-classicality of photon added coherent and thermal radiations. Eur. Phys. J. D
**2006**, 40, 133–138. [Google Scholar] [CrossRef] - Hu, L.-Y.; Fan, H.-Y. Wigner function and density operator of the photon-subtracted squeezed thermal state. Chin. Phys. B
**2009**, 18, 4657–4661. [Google Scholar] [CrossRef] - Bogdanov, Y.I.; Katamadze, K.G.; Avosopiants, G.V.; Belinsky, L.V.; Bogdanova, N.A.; Kalinkin, A.A.; Kulik, S.P. Multiphoton subtracted thermal states: Description, preparation, and reconstruction. Phys. Rev. A
**2017**, 96, 063803. [Google Scholar] [CrossRef][Green Version] - Li, J.; Gröblacher, S.; Zhu, S.Y.; Agarwal, G.S. Generation and detection of non-Gaussian phonon-added coherent states in optomechanical systems. Phys. Rev. A
**2018**, 98, 011801. [Google Scholar] [CrossRef][Green Version] - Fu, H.C.; Sasaki, R. Negative binomial and multinomial states: Probability distributions and coherent states. J. Math. Phys.
**1997**, 38, 3968–3987. [Google Scholar] [CrossRef][Green Version] - Lee Loh, Y.; Kim, M. Visualizing spin states using the spin coherent state representation. Am. J. Phys.
**2015**, 83, 30–35. [Google Scholar] [CrossRef] - Sperling, J.; Walmsley, I.A. Quasiprobability representation of quantum coherence. Phys. Rev. A
**2018**, 97, 062327. [Google Scholar] [CrossRef][Green Version] - Atkins, P.W.; Dobson, J.C.; Coulson, C.A. Angular momentum coherent states. Proc. R. Soc. London. Math. Physical Sci.
**1971**, 321, 321–340. [Google Scholar] [CrossRef] - Zelaya, K.D.; Rosas-Ortiz, O. Optimized Binomial Quantum States of Complex Oscillators with Real Spectrum. J. Phys. Conf. Ser.
**2016**, 698, 12026. [Google Scholar] [CrossRef][Green Version] - Skotiniotis, M.; Gour, G. Alignment of reference frames and an operational interpretation for theG-asymmetry. New J. Phys.
**2012**, 14, 073022. [Google Scholar] [CrossRef][Green Version] - Peruzzo, A.; McClean, J.; Shadbolt, P.; Yung, M.H.; Zhou, X.Q.; Love, P.J.; Aspuru-Guzik, A.; O’Brien, J.L. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun.
**2014**, 5, 4213. [Google Scholar] [CrossRef][Green Version] - Kieferová, M.; Scherer, A.; Berry, D.W. Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series. Phys. Rev. A
**2019**, 99, 042314. [Google Scholar] [CrossRef][Green Version] - Lukacs, E. Characteristic Functions; Griffin Books of Cognate Interest; Hafner Publishing Company: New York, NY, USA, 1970. [Google Scholar]

**Figure 1.**Relation between prepared states and measurements. In the forwards (reverse) process, the state ${\rho}_{SB}^{i}={\rho}_{S}^{i}\otimes {\rho}_{B}^{i}$$\left(\right)$ is prepared, it evolves under U as indicated by the wiggly arrow, and then the measurement ${X}_{SB}^{f}={X}_{S}^{f}\otimes {X}_{B}^{f}$$\left(\right)$ is performed. As indicated by the solid lines, the measurements ${X}_{SB}^{i}$ and ${X}_{SB}^{f}$ are related to the states ${\rho}_{SB}^{i}$ and ${\rho}_{SB}^{f}$, respectively, by the mapping $\mathcal{M}$, defined in Equation (7).

**Figure 2.**Generalised Free Energies. The solid red and dark blue lines show the generalised free energy, $\Delta {F}^{+}$ and $\Delta {F}^{-}$, of the oscillator system for the photon added and photon subtracted equalities, respectively. These are plotted as a function of $\chi =\beta \hslash {\omega}_{i}/2$, the ratio between the initial vacuum fluctuations, $\hslash {\omega}_{i}/2$, and the thermal fluctuations, ${k}_{B}T$, a measure which quantifies the temperature and thus effectively delineates the classical and quantum regimes. The grey dashed line is the usual change in energy $\Delta F$. The dotted lines indicate the contribution of $\Delta {E}_{\mathrm{vac}}$ (purple) and $2\Delta F$ (light blue) to $\Delta {F}^{+}$ and $\Delta {F}^{+}$. In this plot, we suppose $\hslash {\omega}_{f}=1.5\hslash {\omega}_{i}$ and energies are given in units of ${k}_{B}T$.

**Figure 3.**Predicted ratio and $\mathcal{R}$ prefactor. The left figure plots the predicted ratio of the forwards and reverse transition probabilities, i.e., the right hand side of Equation (25), for the photon added (subtracted) Crooks equality as a function of $\chi =\beta \hslash {\omega}_{i}/2$. The right figure plots $\mathcal{R}$ as a function of $\chi $. The red (blue) lines indicates the photon added (subtracted) case and the grey lines indicate the equivalent classical limit. That is, in the left plot the grey line is the right hand side of the classical Crooks equality, Equation (1), and in the right plot the grey line is $\mathcal{R}=1$. The solid lines plot the case $W=2\hslash {\omega}_{i}$ and the dashed lines, $W=0$. Here, we suppose $\hslash {\omega}_{f}=5\hslash {\omega}_{i}$.

**Figure 4.**Quantum distortions of fluctuation relations due to binomial battery states: The left and right plots correspond to ${q}_{\mathrm{align}}$ and ${q}_{\mathrm{size}}$, respectively. The left plot is evaluated for a fixed value ${p}_{f}=0.8$. Both functions are plotted against the quantum-thermodynamic ratio $\chi =\frac{\beta \hslash \omega}{2}$. The plots show that the distortion due to quantum features can both enhance and suppress irreversibility in a process as compared to a “classical equivalent” solely involving energy exchanges. In both cases, we typically find suppressed irreversibility as quantum features dominate for large values of $\chi $. However, when thermodynamic and quantum energy scales are of similar magnitude, we observe unexpected behaviour.

**Figure 5.**Linear optic implementation schematic. A photon added (or subtracted) thermal state is sent into one input arm of a linear optical set up and a coherent state the other. The linear optical set up, consisting of a series of linear optical elements, such as beamsplitters, phase-shifters and mirrors (the particular sequence sketched here is chosen arbitrarily), drives the photonic system and battery with an energy conserving and time reversal invariant operation. Finally, a coherent state measurement is performed on one output arm of the optical setup using a homodyne detection and the number of photons out put is measured in the other arm.

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**MDPI and ACS Style**

Holmes, Z.; Hinds Mingo, E.; Chen, C.Y.-R.; Mintert, F.
Quantifying Athermality and Quantum Induced Deviations from Classical Fluctuation Relations. *Entropy* **2020**, *22*, 111.
https://doi.org/10.3390/e22010111

**AMA Style**

Holmes Z, Hinds Mingo E, Chen CY-R, Mintert F.
Quantifying Athermality and Quantum Induced Deviations from Classical Fluctuation Relations. *Entropy*. 2020; 22(1):111.
https://doi.org/10.3390/e22010111

**Chicago/Turabian Style**

Holmes, Zoë, Erick Hinds Mingo, Calvin Y.-R. Chen, and Florian Mintert.
2020. "Quantifying Athermality and Quantum Induced Deviations from Classical Fluctuation Relations" *Entropy* 22, no. 1: 111.
https://doi.org/10.3390/e22010111