# Effect of Quantum Coherence on Landauer’s Principle

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## Abstract

**:**

## 1. Introduction

## 2. Thermal Quantum Information Erasure

- the protocol involves an information-bearing system S and a thermal reservoir R, both described by certain Hamiltonians, denoted ${H}_{S}$ and ${H}_{R}$, respectively,
- the reservoir R is initially in the thermal equilibrium with a certain inverse temperature $\beta $, ${\rho}_{R}\left(0\right)={\rho}_{R}^{\mathrm{eq}}\equiv \mathrm{exp}(-\beta {H}_{R})/{\mathrm{Tr}}_{R}\left[\mathrm{exp}(-\beta {H}_{R})\right]$, where ${\rho}_{R}\left(t\right)$ is the reduced density operator of R,
- the system S and the reservoir R are initially uncorrelated, ${\rho}_{\mathrm{tot}}\left(0\right)={\rho}_{S}\left(0\right)\otimes {\rho}_{R}^{\mathrm{eq}}$, where ${\rho}_{\mathrm{tot}}\left(0\right)$ is the total density operator of S+R and ${\rho}_{S}\left(t\right)$ is the reduced density operator of S,
- the erasure process itself proceeds by a unitary evolution generated by the total Hamiltonian $H={H}_{\mathrm{S}}+{H}_{R}+{H}_{SR}$, where ${H}_{SR}$ is an interaction between S and R.

## 3. Lower Bounds for the Energy Dissipation

#### 3.1. Entropic Bound

#### 3.2. Thermodynamic Bound

## 4. Full-Counting Statistics Formalism

## 5. Spin—Boson Model

#### 5.1. Model

#### 5.2. The Bloch Vector Representation

## 6. Relative Tightness of the Bounds

#### 6.1. Dependence on Initial State

#### 6.2. Dependence on Quantum Coherence

## 7. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. The Bloch Equation Including the Counting Field

## Appendix B. A Single Spin Subjected to a Tilted Magnetic Field

## References

- Landauer, R. Irreversibility and Heat Generation in the Computing Process. IBM J. Res. Dev.
**1961**, 5, 183. [Google Scholar] [CrossRef] - Penrose, O. Foundations of Statistical Mechanics: A Deductive Treatment; Pergamon: New York, NY, USA, 1970. [Google Scholar]
- Bennet, C.H. Logical Reversibility of Computation. IBM J. Res. Dev.
**1973**, 17, 525. [Google Scholar] [CrossRef] - Landauer, R. Information is Physical. Phys. Today
**1991**, 44, 23. [Google Scholar] [CrossRef] - Plenio, M.B.; Vitelli, V. The physics of forgetting: Landauer’s erasure principle and information theory. Contemp. Phys.
**2001**, 42, 25. [Google Scholar] [CrossRef] [Green Version] - Shizume, K. Heat generation required by information erasure. Phys. Rev. E
**1995**, 52, 3495. [Google Scholar] [CrossRef] - Piechocinska, B. Information erasure. Phys. Rev. A
**2000**, 61, 062314. [Google Scholar] [CrossRef] - Toyabe, S.; Sagawa, T.; Ueda, M.; Muneyuki, E.; Sano, M. Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nat. Phys.
**2010**, 6, 988. [Google Scholar] [CrossRef] [Green Version] - Orlov, A.O.; Lent, C.S.; Thorpe, C.C.; Boechler, G.P.; Snider, G.L. Experimental Test of Landauer’s Principle at the Sub-k
_{B}T Level. Jpn. J. Appl. Phys.**2012**, 51, 06FE10. [Google Scholar] [CrossRef] - Bérut, A.; Arakelyan, A.; Petrosyan, A.; Ciliberto, S.; Dillenschneider, R.; Lutz, E. Experimental verification of Landauer’s principle linking information and thermodynamics. Nature
**2012**, 483, 187. [Google Scholar] [CrossRef] - Jun, Y.; Gavrilov, M.; Bechhoefer, J. High-Precision Test of Landauer’s Principle in a Feedback Trap. Phys. Rev. Lett.
**2014**, 113, 190601. [Google Scholar] [CrossRef] [Green Version] - Hilt, S.; Shabbir, S.; Anders, J.; Lutz, E. Landauer’s principle in the quantum regime. Phys. Rev. E
**2011**, 83, 030102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Reeb, D.; Wolf, M.M. An improved Landauer principle with finite-size corrections. New J. Phys.
**2014**, 16, 103011. [Google Scholar] [CrossRef] - Sagawa, T.; Ueda, M. Minimal Energy Cost for Thermodynamic Information Processing: Measurement and Information Erasure. Phys. Rev. Lett.
**2009**, 102, 250602. [Google Scholar] [CrossRef] [Green Version] - Faist, P.; Dupuis, F.; Oppenheim, J.; Renner, R. The minimal work cost of information processing. Nat. Commun.
**2015**, 6, 7669. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mohammady, M.H.; Mohseni, M.; Omar, Y. Minimising the heat dissipation of quantum information erasure. New J. Phys.
**2016**, 18, 015011. [Google Scholar] [CrossRef] [Green Version] - Bedingham, D.J.; Maroney, O.J.E. The thermodynamic cost of quantum operations. New J. Phys.
**2016**, 18, 113050. [Google Scholar] [CrossRef] [Green Version] - Peterson, J.P.S.; Sarthour, R.S.; Souza, A.M.; Oliveira, I.S.; Goold, J.; Modi, K.; Soares-Pinto, D.O.; Céleri, L.C. Experimental demonstration of information to energy conversion in a quantum system at the Landauer limit. Proc. R. Soc. A
**2016**, 472, 20150813. [Google Scholar] [CrossRef] [Green Version] - Chitambar, E.; Gour, G. Quantum resource theories. Rev. Mod. Phys.
**2019**, 91, 025001. [Google Scholar] [CrossRef] [Green Version] - Goold, J.; Huber, M.; Riera, A.; del Rio, L.; Skrzypczyk, P. The role of quantum information in thermodynamics—A topical review. J. Phys. A Math. Theor.
**2016**, 49, 143001. [Google Scholar] [CrossRef] - Millen, J.; Xuereb, A. Perspective on quantum thermodynamics. New J. Phys.
**2016**, 18, 011002. [Google Scholar] [CrossRef] - Goold, J.; Paternostro, M.; Modi, K. Nonequilibrium Quantum Landauer Principle. Phys. Rev. Lett.
**2015**, 114, 060602. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Guarnieri, G.; Campbell, S.; Goold, J.; Pigeon, S.; Vacchini, B.; Paternostro, M. Full counting statistics approach to the quantum non-equilibrium Landauer bound. New J. Phys.
**2017**, 19, 103038. [Google Scholar] [CrossRef] [Green Version] - Campbell, S.; Guarnieri, G.; Paternostro, M.; Vacchini, B. Nonequilibrium quantum bounds to Landauer’s principle: Tightness and effectiveness. Phys. Rev. A
**2017**, 96, 042109. [Google Scholar] [CrossRef] [Green Version] - Hashimoto, K.; Vacchini, B.; Uchiyama, C. Lower bounds for the mean dissipated heat in an open quantum system. Phys. Rev. A
**2020**, 101, 052114. [Google Scholar] [CrossRef] - Miller, H.J.D.; Guarnieri, G.; Mitchison, M.T.; Goold, J. Quantum Fluctuations Hinder Finite-Time Information Erasure near the Landauer Limit. Phys. Rev. Lett.
**2020**, 125, 160602. [Google Scholar] [CrossRef] [PubMed] - Vu, T.V.; Saito, S. Finite-Time Quantum Landauer Principle and Quantum Coherence. Phys. Rev. Lett.
**2022**, 128, 010602. [Google Scholar] - Uchiyama, C. Nonadiabatic effect on the quantum heat flux control. Phys. Rev. E
**2014**, 89, 052108. [Google Scholar] [CrossRef] [Green Version] - Guarnieri, G.; Uchiyama, C.; Vacchini, B. Energy backflow and non-Markovian dynamics. Phys. Rev. A
**2016**, 93, 012118. [Google Scholar] [CrossRef] [Green Version] - Hashimoto, K.; Uchiyama, C. Nonadiabaticity in Quantum Pumping Phenomena under Relaxation. Entropy
**2019**, 21, 842. [Google Scholar] [CrossRef] [Green Version] - Esposito, M.; Lindenberg, K.; Van den Broeck, C. Entropy production as correlation between system and reservoir. New J. Phys.
**2010**, 12, 013013. [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. [Google Scholar] [CrossRef] [Green Version] - Kubo, R. Stochastic Liouville Equations. J. Math. Phys.
**1963**, 4, 174. [Google Scholar] [CrossRef] - Van Kampen, N.G. A cumulant expansion for stochastic linear differential equations. I. Physica
**1974**, 74, 215. [Google Scholar] [CrossRef] - Van Kampen, N.G. A cumulant expansion for stochastic linear differential equations. II. Physica
**1974**, 74, 239. [Google Scholar] [CrossRef] - Hashitsume, N.; Shibata, F.; Shingu, M. Quantal master equation valid for any time scale. J. Stat. Phys.
**1977**, 17, 155. [Google Scholar] [CrossRef] - Shibata, F.; Takahashi, Y.; Hashitsume, N. A generalized stochastic liouville equation. Non-Markovian versus memoryless master equations. J. Stat. Phys.
**1977**, 17, 171. [Google Scholar] [CrossRef] - Chaturvedi, S.; Shibata, F. Time-convolutionless projection operator formalism for elimination of fast variables. Applications to Brownian motion. Z. Phys. B Condens. Matter
**1979**, 35, 297. [Google Scholar] [CrossRef] - Shibata, F.; Arimitsu, T. Expansion Formulas in Nonequilibrium Statistical Mechanics. J. Phys. Soc. Jpn.
**1980**, 49, 891. [Google Scholar] [CrossRef] - Uchiyama, C.; Shibata, F. Unified projection operator formalism in nonequilibrium statistical mechanics. Phys. Rev. E
**1999**, 60, 2636. [Google Scholar] [CrossRef] - Breuer, H.-P.; Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press: Oxford, UK, 2002. [Google Scholar]
- Shirai, Y.; Hashimoto, K.; Tezuka, R.; Uchiyama, C.; Hatano, N. Non-Markovian effect on quantum Otto engine: Role of system-reservoir interaction. Phys. Rev. Res.
**2021**, 3, 023078. [Google Scholar] [CrossRef] - Breuer, H.-P.; Burgarth, D.; Petruccione, F. Non-Markovian dynamics in a spin star system: Exact solution and approximation techniques. Phys. Rev. B
**2004**, 70, 045323. [Google Scholar] [CrossRef] [Green Version] - Cucchietti, F.M.; Paz, J.P.; Zurek, W.H. Decoherence from spin environments. Phys. Rev. A
**2005**, 72, 052113. [Google Scholar] [CrossRef] [Green Version] - Camalet, S.; Chitra, R. Effect of random interactions in spin baths on decoherence. Phys. Rev. B
**2007**, 75, 094434. [Google Scholar] [CrossRef] [Green Version] - Segal, D. Two-level system in spin baths: Non-adiabatic dynamics and heat transport. J. Chem. Phys.
**2014**, 140, 164110. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mirza, A.R.; Zia, M.; Chaudhry, A.Z. Master equation incorporating the system-environment correlations present in the joint equilibrium state. Phys. Rev. A
**2021**, 104, 042205. [Google Scholar] [CrossRef] - Taylor, J.M.; Marcus, C.M.; Lukin, M.D. Long-Lived Memory for Mesoscopic Quantum Bits. Phys. Rev. Lett.
**2003**, 90, 206803. [Google Scholar] [CrossRef] - Wu, L.-A. Dressed qubits in nuclear spin baths. Phys. Rev. A
**2010**, 81, 044305. [Google Scholar] [CrossRef] [Green Version] - Jing, J.; Wu, L.-A. Decoherence and control of a qubit in spin baths: An exact master equation study. Sci. Rep.
**2018**, 8, 1471. [Google Scholar] [CrossRef] [Green Version] - Ivády, V. Longitudinal spin relaxation model applied to point-defect qubit systems. Phys. Rev. B
**2020**, 101, 155203. [Google Scholar] [CrossRef] [Green Version] - Kwiatkowski, D.; Szańkowski, P.; Cywiński, Ł. Influence of nuclear spin polarization on the spin-echo signal of an NV-center qubit. Phys. Rev. B
**2020**, 101, 155412. [Google Scholar] [CrossRef] [Green Version] - Vaccaro, J.A.; Barnett, S.M. Information erasure without an energy cost. Proc. R. Soc. A
**2011**, 467, 1770. [Google Scholar] [CrossRef] [Green Version] - Croucher, T.; Vaccaro, J.A. Thermodynamics of memory erasure via a spin reservoir. Phys. Rev. E
**2021**, 103, 042140. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Dependences of the energy $\langle \Delta Q\rangle $ and the bounds ${\mathcal{B}}_{T,E}$ on the initial state of the system for $\theta =\pi /4$. The initial condition is chosen by changing ${v}_{x}^{\left(0\right)}\left(0\right)$ and ${v}_{z}^{\left(0\right)}\left(0\right)$ while fixing ${v}_{y}\left(0\right)=0$. (

**a**) 3D plot of $\langle \Delta Q\rangle $ (orange surface), ${\mathcal{B}}_{T}$ (blue surface) and ${\mathcal{B}}_{E}$ (red surface) with respect to ${v}_{x}\left(0\right)$ and ${v}_{z}\left(0\right)$. The purple circle indicates the surface of Bloch sphere with ${v}_{y}^{\left(0\right)}\left(0\right)=0$. (

**b**) cross-section of the 3D plot at ${v}_{x}^{\left(0\right)}\left(0\right)=0$ plotted with respect to ${v}_{z}^{\left(0\right)}\left(0\right)$. (

**c**) cross-section at ${v}_{z}^{\left(0\right)}\left(0\right)=0$ plotted with respect to ${v}_{x}^{\left(0\right)}\left(0\right)$. For the numerical calculations, we set the parameters to $\lambda =0.01$, $\Omega =1$, and $\beta =1$.

**Figure 2.**Dependences of the energy $\langle \Delta Q\rangle $ and the bounds ${\mathcal{B}}_{T,E}$ on the initial state of the system for $\theta =\pi /2$. (

**a**) 3D plot of $\langle \Delta Q\rangle $ (orange surface), ${\mathcal{B}}_{T}$ (blue surface), and ${\mathcal{B}}_{E}$ (red surface) with respect to ${v}_{x}\left(0\right)$ and ${v}_{z}\left(0\right)$. (

**b**) cross-section of the 3D plot at ${v}_{x}^{\left(0\right)}\left(0\right)=0$ plotted with respect to ${v}_{z}^{\left(0\right)}\left(0\right)=0$. (

**c**) cross-section at ${v}_{z}^{\left(0\right)}\left(0\right)=0$ plotted with respect to ${v}_{x}^{\left(0\right)}\left(0\right)=0$. For the numerical calculations, we set the parameters to $\lambda =0.01$, $\Omega =1$, and $\beta =1$ (same as in Figure 1).

**Figure 3.**Dependences of the energy $\langle \Delta Q\rangle $ and the bounds ${\mathcal{B}}_{T,E}$ on the coherence parameter $\theta $ and the initial population ${v}_{z}^{\left(0\right)}\left(0\right)$ with setting ${v}_{x}^{\left(0\right)}\left(0\right)={v}_{y}^{\left(0\right)}\left(0\right)=0$. Panel (

**a**) shows a 3D plot of $\langle \Delta Q\rangle $ (orange surface), ${\mathcal{B}}_{T}$ (blue surface) and ${\mathcal{B}}_{E}$ (red surface). Panels (

**b**,

**c**): cross-sections of the 3D plot for two pure initial states (

**b**) ${v}_{z}^{\left(0\right)}\left(0\right)=1$ and (

**c**) ${v}_{z}^{\left(0\right)}\left(0\right)=-1$. Panels (

**d**,

**e**): cross-sections of the 3D plot corresponding to thermal initial states ${\rho}_{S}^{\left(0\right)}\left(0\right)=\mathrm{exp}[-{\beta}_{S}{H}_{S}]/{\mathrm{Tr}}_{S}\left[\mathrm{exp}[-{\beta}_{S}{H}_{S}]\right]$ with (

**d**) ${\beta}_{S}=0$ (high temperature limit) and (

**e**) ${\beta}_{S}=1(=\beta )$. For the numerical calculations, we set the parameters to $\lambda =0.01$, $\Omega =1$, and $\beta =1$ (same as in Figure 1).

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Hashimoto, K.; Uchiyama, C.
Effect of Quantum Coherence on Landauer’s Principle. *Entropy* **2022**, *24*, 548.
https://doi.org/10.3390/e24040548

**AMA Style**

Hashimoto K, Uchiyama C.
Effect of Quantum Coherence on Landauer’s Principle. *Entropy*. 2022; 24(4):548.
https://doi.org/10.3390/e24040548

**Chicago/Turabian Style**

Hashimoto, Kazunari, and Chikako Uchiyama.
2022. "Effect of Quantum Coherence on Landauer’s Principle" *Entropy* 24, no. 4: 548.
https://doi.org/10.3390/e24040548