Energy storage and coherence in closed and open quantum batteries

Francesco Caravelli1, Bin Yan1,2, Luis Pedro García-Pintos3, and Alioscia Hamma4

1Theoretical Division (T4), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
2Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
3Joint Center for Quantum Information and Computer Science and Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
4Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA

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Abstract

We study the role of coherence in closed and open quantum batteries. We obtain upper bounds to the work performed or energy exchanged by both closed and open quantum batteries in terms of coherence. Specifically, we show that the energy storage can be bounded by the Hilbert-Schmidt coherence of the density matrix in the spectral basis of the unitary operator that encodes the evolution of the battery. We also show that an analogous bound can be obtained in terms of the battery's Hamiltonian coherence in the basis of the unitary operator by evaluating their commutator. We apply these bounds to a 4-state quantum system and the anisotropic XY Ising model in the closed system case, and the Spin-Boson model in the open case.

We obtain upper bounds to the work performed or energy exchanged by both closed and open quantum batteries in terms of coherence.

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[1] Johan Aberg. Truly work-like work extraction via a single-shot analysis. Nat. Commun., 4: 1925, 2013. ISSN 2041-1723. 10.1038/​ncomms2712.
https:/​/​doi.org/​10.1038/​ncomms2712

[2] Johan Åberg. Catalytic coherence. Phys. Rev. Lett., 113 (15): 150402, October 2014. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.113.150402.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.150402

[3] Álvaro M Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry. Fluctuating work: From quantum thermodynamical identities to a second law equality. Phys. Rev. X, 6 (4): 041017, October 2016. 10.1103/​PhysRevX.6.041017.
https:/​/​doi.org/​10.1103/​PhysRevX.6.041017

[4] Álvaro M Alhambra, Georgios Styliaris, Nayeli A Rodríguez-Briones, Jamie Sikora, and Eduardo Martín-Martínez. Fundamental limitations to local energy extraction in quantum systems. Phys. Rev. Lett., 123 (19): 190601, November 2019. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.123.190601.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.190601

[5] R Alicki. The quantum open system as a model of the heat engine. J. Phys. A Math. Gen., 12 (5): L103, May 1979. ISSN 0305-4470. 10.1088/​0305-4470/​12/​5/​007.
https:/​/​doi.org/​10.1088/​0305-4470/​12/​5/​007

[6] Robert Alicki and Mark Fannes. Entanglement boost for extractable work from ensembles of quantum batteries. Phys. Rev. E, 87 (4): 042123, April 2013. 10.1103/​PhysRevE.87.042123.
https:/​/​doi.org/​10.1103/​PhysRevE.87.042123

[7] Robert Alicki and David Gelbwaser-Klimovsky. Non-equilibrium quantum heat machines. New J. Phys., 17 (11): 115012, November 2015. ISSN 1367-2630. 10.1088/​1367-2630/​17/​11/​115012.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​11/​115012

[8] Robert Alicki, Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Thermodynamics of quantum information systems — hamiltonian description. Open Syst. Inf. Dyn., 11 (3): 205–217, September 2004. ISSN 1230-1612, 1573-1324. 10.1023/​B:OPSY.0000047566.72717.71.
https:/​/​doi.org/​10.1023/​B:OPSY.0000047566.72717.71

[9] A E Allahverdyan, R Balian, and Th M Nieuwenhuizen. Maximal work extraction from finite quantum systems. EPL, 67 (4): 565, August 2004a. ISSN 0295-5075. 10.1209/​epl/​i2004-10101-2.
https:/​/​doi.org/​10.1209/​epl/​i2004-10101-2

[10] A E Allahverdyan, R Balian, and Th M Nieuwenhuizen. Maximal work extraction from finite quantum systems. EPL, 67 (4): 565, August 2004b. ISSN 0295-5075. 10.1209/​epl/​i2004-10101-2.
https:/​/​doi.org/​10.1209/​epl/​i2004-10101-2

[11] Janet Anders and Massimiliano Esposito. Focus on quantum thermodynamics. New J. Phys., 19 (1): 010201, January 2017. ISSN 1367-2630. 10.1088/​1367-2630/​19/​1/​010201.
https:/​/​doi.org/​10.1088/​1367-2630/​19/​1/​010201

[12] Gian Marcello Andolina, Donato Farina, Andrea Mari, Vittorio Pellegrini, Vittorio Giovannetti, and Marco Polini. Charger-mediated energy transfer in exactly solvable models for quantum batteries. Phys. Rev. B Condens. Matter, 98 (20): 205423, November 2018. ISSN 0163-1829. 10.1103/​PhysRevB.98.205423.
https:/​/​doi.org/​10.1103/​PhysRevB.98.205423

[13] Gian Marcello Andolina, Maximilian Keck, Andrea Mari, Michele Campisi, Vittorio Giovannetti, and Marco Polini. Extractable work, the role of correlations, and asymptotic freedom in quantum batteries. Phys. Rev. Lett., 122 (4): 047702, February 2019a. ISSN 0031-9007. 10.1103/​PhysRevLett.122.047702.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.047702

[14] Gian Marcello Andolina, Maximilian Keck, Andrea Mari, Vittorio Giovannetti, and Marco Polini. Quantum versus classical many-body batteries. Phys. Rev. B Condens. Matter, 99 (20): 205437, May 2019b. ISSN 0163-1829. 10.1103/​PhysRevB.99.205437.
https:/​/​doi.org/​10.1103/​PhysRevB.99.205437

[15] V G Bagrov, D M Gitman, M C Baldiotti, and A D Levin. Spin equation and its solutions. Ann. Phys., 14 (11-12): 764–789, December 2005. ISSN 0003-3804, 1521-3889. 10.1002/​andp.200510176.
https:/​/​doi.org/​10.1002/​andp.200510176

[16] V G Bagrov, M C Baldiotti, D M Gitman, and A D Levin. Two interacting spins in external fields. four-level systems. Ann. Phys., 16 (4): 274–285, April 2007. ISSN 0003-3804, 1521-3889. 10.1002/​andp.200610231.
https:/​/​doi.org/​10.1002/​andp.200610231

[17] T Baumgratz, M Cramer, and M B Plenio. Quantifying coherence. Phys. Rev. Lett., 113 (14): 140401, September 2014. ISSN 0031-9007. 10.1103/​PhysRevLett.113.140401.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.140401

[18] Francis A Bayocboc and Francis N C Paraan. Exact work statistics of quantum quenches in the anisotropic $XY$ model. Phys. Rev. E, 92 (3): 032142, September 2015. 10.1103/​PhysRevE.92.032142.
https:/​/​doi.org/​10.1103/​PhysRevE.92.032142

[19] Manabendra N Bera, Arnau Riera, Maciej Lewenstein, and Andreas Winter. Generalized laws of thermodynamics in the presence of correlations. Nat. Commun., 8 (1): 1–6, December 2017. ISSN 2041-1723, 2041-1723. 10.1038/​s41467-017-02370-x.
https:/​/​doi.org/​10.1038/​s41467-017-02370-x

[20] Manabendra Nath Bera, Arnau Riera, Maciej Lewenstein, Zahra Baghali Khanian, and Andreas Winter. Thermodynamics as a consequence of information conservation. Quantum, 3 (121): 121, February 2019. ISSN 2521-327X. 10.22331/​q-2019-02-14-121.
https:/​/​doi.org/​10.22331/​q-2019-02-14-121

[21] Sourav Bhattacharjee and Amit Dutta. Quantum thermal machines and batteries. August 2020. https:/​/​arxiv.org/​abs/​2008.07889.
arXiv:2008.07889

[22] Arpan Bhattacharyya, Wissam Chemissany, S Shajidul Haque, and Bin Yan. Towards the web of quantum chaos diagnostics. September 2019. https:/​/​arxiv.org/​abs/​1909.01894.
arXiv:1909.01894

[23] Felix C Binder, Sai Vinjanampathy, Kavan Modi, and John Goold. Quantacell: powerful charging of quantum batteries. New J. Phys., 17 (7): 075015, July 2015. ISSN 1367-2630. 10.1088/​1367-2630/​17/​7/​075015.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​7/​075015

[24] Albrecht Böttcher and David Wenzel. The frobenius norm and the commutator. Linear Algebra Appl., 429 (8): 1864–1885, October 2008. ISSN 0024-3795. 10.1016/​j.laa.2008.05.020.
https:/​/​doi.org/​10.1016/​j.laa.2008.05.020

[25] Fernando Brandão, Michał Horodecki, Nelly Ng, Jonathan Oppenheim, and Stephanie Wehner. The second laws of quantum thermodynamics. Proc. Natl. Acad. Sci. U. S. A., 112 (11): 3275–3279, March 2015. ISSN 0027-8424, 1091-6490. 10.1073/​pnas.1411728112.
https:/​/​doi.org/​10.1073/​pnas.1411728112

[26] Fernando G S L Brandão, Michał Horodecki, Jonathan Oppenheim, Joseph M Renes, and Robert W Spekkens. Resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett., 111 (25): 250404, December 2013. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.111.250404.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.250404

[27] Kay Brandner, Michael Bauer, Michael T Schmid, and Udo Seifert. Coherence-enhanced efficiency of feedback-driven quantum engines. New J. Phys., 17 (6): 065006, June 2015. ISSN 1367-2630. 10.1088/​1367-2630/​17/​6/​065006.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​6/​065006

[28] B Çakmak, A Manatuly, and Ö E Müstecaplıoğlu. Thermal production, protection, and heat exchange of quantum coherences. Phys. Rev. A, 96 (3): 032117, September 2017. ISSN 1050-2947. 10.1103/​PhysRevA.96.032117.
https:/​/​doi.org/​10.1103/​PhysRevA.96.032117

[29] Barış Çakmak. Ergotropy from coherences in an open quantum system. Phys Rev E, 102 (4-1): 042111, October 2020. ISSN 2470-0053, 2470-0045. 10.1103/​PhysRevE.102.042111.
https:/​/​doi.org/​10.1103/​PhysRevE.102.042111

[30] A O Caldeira and A J Leggett. Quantum tunnelling in a dissipative system. Ann. Phys., 149 (2): 374–456, September 1983. ISSN 0003-4916. 10.1016/​0003-4916(83)90202-6.
https:/​/​doi.org/​10.1016/​0003-4916(83)90202-6

[31] F Campaioli, F. A. Pollock, and S Vinjanampathy. Quantum Batteries. In: Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions. Springer, Cham, 2018. 10.1007/​978-3-319-99046-0.
https:/​/​doi.org/​10.1007/​978-3-319-99046-0

[32] Francesco Campaioli, Felix A Pollock, Felix C Binder, Lucas Céleri, John Goold, Sai Vinjanampathy, and Kavan Modi. Enhancing the charging power of quantum batteries. Phys. Rev. Lett., 118 (15): 150601, April 2017. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.118.150601.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.150601

[33] Francesco Caravelli, Ghislaine Coulter-De Wit, Luis Pedro García-Pintos, and Alioscia Hamma. Random quantum batteries. Phys. Rev. Research, 2 (2): 023095, April 2020. 10.1103/​PhysRevResearch.2.023095.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.023095

[34] M Carrega, A Crescente, D Ferraro, and M Sassetti. Dissipative dynamics of an open quantum battery. New J. Phys., 22 (8): 083085, August 2020. ISSN 1367-2630. 10.1088/​1367-2630/​abaa01.
https:/​/​doi.org/​10.1088/​1367-2630/​abaa01

[35] A Chenu, I L Egusquiza, J Molina-Vilaplana, and A Del Campo. Quantum work statistics, loschmidt echo and information scrambling. Sci. Rep., 8 (1): 12634, August 2018. ISSN 2045-2322. 10.1038/​s41598-018-30982-w.
https:/​/​doi.org/​10.1038/​s41598-018-30982-w

[36] Aurélia Chenu, Javier Molina-Vilaplana, and Adolfo del Campo. Work statistics, loschmidt echo and information scrambling in chaotic quantum systems. Quantum, 3: 127, March 2019. ISSN 2521-327X. 10.22331/​q-2019-03-04-127.
https:/​/​doi.org/​10.22331/​q-2019-03-04-127

[37] Luis A Correa, José P Palao, Gerardo Adesso, and Daniel Alonso. Performance bound for quantum absorption refrigerators. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 87 (4): 042131, April 2013. ISSN 1539-3755, 1550-2376. 10.1103/​PhysRevE.87.042131.
https:/​/​doi.org/​10.1103/​PhysRevE.87.042131

[38] Piotr Ć wikliński, Michał Studziński, Michał Horodecki, and Jonathan Oppenheim. Limitations on the evolution of quantum coherences: Towards fully quantum second laws of thermodynamics. Phys. Rev. Lett., 115 (21): 210403, November 2015. ISSN 0031-9007. 10.1103/​PhysRevLett.115.210403.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.210403

[39] H De Readt, B De Raedt, and A Lagendijk. Thermodynamics of the two-dimensional spin-1/​2 XY model. Z. Phys. B: Condens. Matter, 57 (3): 209–220, September 1984. ISSN 1431-584X. 10.1007/​BF01318413.
https:/​/​doi.org/​10.1007/​BF01318413

[40] A del Campo, J Goold, and M Paternostro. More bang for your buck: super-adiabatic quantum engines. Sci. Rep., 4: 6208, August 2014. ISSN 2045-2322. 10.1038/​srep06208.
https:/​/​doi.org/​10.1038/​srep06208

[41] Dario Ferraro, Michele Campisi, Gian Marcello Andolina, Vittorio Pellegrini, and Marco Polini. High-Power collective charging of a Solid-State quantum battery. Phys. Rev. Lett., 120 (11): 117702, March 2018. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.120.117702.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.117702

[42] R P Feynman and F L Vernon. The theory of a general quantum system interacting with a linear dissipative system. Ann. Phys., 24: 118–173, October 1963. ISSN 0003-4916. 10.1016/​0003-4916(63)90068-X.
https:/​/​doi.org/​10.1016/​0003-4916(63)90068-X

[43] G Francica, J Goold, and F Plastina. Role of coherence in the nonequilibrium thermodynamics of quantum systems. Phys. Rev. E, 99 (4): 042105, April 2019. 10.1103/​PhysRevE.99.042105.
https:/​/​doi.org/​10.1103/​PhysRevE.99.042105

[44] G Francica, F C Binder, G Guarnieri, M T Mitchison, J Goold, and F Plastina. Quantum coherence and ergotropy. Phys. Rev. Lett., 125 (18): 180603, October 2020. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.125.180603.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.180603

[45] Max F Frenzel, David Jennings, and Terry Rudolph. Reexamination of pure qubit work extraction. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 90 (5-1): 052136, November 2014. ISSN 1539-3755, 1550-2376. 10.1103/​PhysRevE.90.052136.
https:/​/​doi.org/​10.1103/​PhysRevE.90.052136

[46] Nicolai Friis and Marcus Huber. Precision and work fluctuations in gaussian battery charging. Quantum, 2 (61): 61, April 2018. ISSN 2521-327X. 10.22331/​q-2018-04-23-61.
https:/​/​doi.org/​10.22331/​q-2018-04-23-61

[47] Ken Funo, Jing-Ning Zhang, Cyril Chatou, Kihwan Kim, Masahito Ueda, and Adolfo Del Campo. Universal work fluctuations during shortcuts to adiabaticity by counterdiabatic driving. Phys. Rev. Lett., 118 (10): 100602, March 2017. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.118.100602.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.100602

[48] R Gallego, J Eisert, and H Wilming. Thermodynamic work from operational principles. New J. Phys., 18 (10): 103017, October 2016. ISSN 1367-2630. 10.1088/​1367-2630/​18/​10/​103017.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​10/​103017

[49] Luis Pedro García-Pintos, Alioscia Hamma, and Adolfo Del Campo. Fluctuations in extractable work bound the charging power of quantum batteries. Phys. Rev. Lett., 125 (4): 040601, July 2020. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.125.040601.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.040601

[50] John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, and Paul Skrzypczyk. The role of quantum information in thermodynamics—a topical review. J. Phys. A: Math. Theor., 49 (14): 143001, February 2016. ISSN 1751-8121. 10.1088/​1751-8113/​49/​14/​143001.
https:/​/​doi.org/​10.1088/​1751-8113/​49/​14/​143001

[51] Gilad Gour, David Jennings, Francesco Buscemi, Runyao Duan, and Iman Marvian. Quantum majorization and a complete set of entropic conditions for quantum thermodynamics. Nat. Commun., 9 (1): 1–9, December 2018. ISSN 2041-1723, 2041-1723. 10.1038/​s41467-018-06261-7.
https:/​/​doi.org/​10.1038/​s41467-018-06261-7

[52] Alioscia Hamma and Paolo Zanardi. Quantum entangling power of adiabatically connected hamiltonians. Phys. Rev. A, 69 (6): 062319, June 2004. ISSN 1050-2947. 10.1103/​PhysRevA.69.062319.
https:/​/​doi.org/​10.1103/​PhysRevA.69.062319

[53] Alioscia Hamma, Siddhartha Santra, and Paolo Zanardi. Quantum entanglement in random physical states. Phys. Rev. Lett., 109 (4): 040502, July 2012a. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.109.040502.
https:/​/​doi.org/​10.1103/​PhysRevLett.109.040502

[54] Alioscia Hamma, Siddhartha Santra, and Paolo Zanardi. Ensembles of physical states and random quantum circuits on graphs. Phys. Rev. A, 86 (5): 052324, November 2012b. ISSN 1050-2947. 10.1103/​PhysRevA.86.052324.
https:/​/​doi.org/​10.1103/​PhysRevA.86.052324

[55] Alioscia Hamma, Georgios Styliaris, and Paolo Zanardi. Localizable quantum coherence. Phys. Lett. A, 397: 127264, May 2021. ISSN 0375-9601. 10.1016/​j.physleta.2021.127264.
https:/​/​doi.org/​10.1016/​j.physleta.2021.127264

[56] Timothy F Havel. Robust procedures for converting among lindblad, kraus and matrix representations of quantum dynamical semigroups. J. Math. Phys., 44 (2): 534–557, February 2003. ISSN 0022-2488. 10.1063/​1.1518555.
https:/​/​doi.org/​10.1063/​1.1518555

[57] Roger A Horn and Charles R Johnson. Matrix Analysis. Cambridge University Press, 1999. ISBN 9780521305877. 10.1017/​CBO9780511840371.
https:/​/​doi.org/​10.1017/​CBO9780511840371

[58] Michał Horodecki and Jonathan Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nat. Commun., 4 (1): 1–6, June 2013. ISSN 2041-1723, 2041-1723. 10.1038/​ncomms3059.
https:/​/​doi.org/​10.1038/​ncomms3059

[59] R Islam, E E Edwards, K Kim, S Korenblit, C Noh, H Carmichael, G-D Lin, L-M Duan, C-C Joseph Wang, J K Freericks, and C Monroe. Onset of a quantum phase transition with a trapped ion quantum simulator. Nat. Commun., 2 (1): 1–6, July 2011. ISSN 2041-1723, 2041-1723. 10.1038/​ncomms1374.
https:/​/​doi.org/​10.1038/​ncomms1374

[60] D Janzing, P Wocjan, R Zeier, R Geiss, and Th Beth. Thermodynamic cost of reliability and low temperatures: Tightening landauer's principle and the second law. Int. J. Theor. Phys., 39 (12): 2717–2753, December 2000. ISSN 1572-9575. 10.1023/​A:1026422630734.
https:/​/​doi.org/​10.1023/​A:1026422630734

[61] Sergi Julià-Farré, Tymoteusz Salamon, Arnau Riera, Manabendra N Bera, and Maciej Lewenstein. Bounds on the capacity and power of quantum batteries. Phys. Rev. Research, 2 (2): 023113, May 2020. 10.1103/​PhysRevResearch.2.023113.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.023113

[62] E Knill, R Laflamme, and L Viola. Theory of quantum error correction for general noise. Phys. Rev. Lett., 84 (11): 2525–2528, March 2000. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.84.2525.
https:/​/​doi.org/​10.1103/​PhysRevLett.84.2525

[63] Kamil Korzekwa, Matteo Lostaglio, Jonathan Oppenheim, and David Jennings. The extraction of work from quantum coherence. New J. Phys., 18 (2): 023045, February 2016. ISSN 1367-2630. 10.1088/​1367-2630/​18/​2/​023045.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023045

[64] Ronnie Kosloff. A quantum mechanical open system as a model of a heat engine. J. Chem. Phys., 80 (4): 1625–1631, February 1984. ISSN 0021-9606. 10.1063/​1.446862.
https:/​/​doi.org/​10.1063/​1.446862

[65] Ronnie Kosloff and Amikam Levy. Quantum heat engines and refrigerators: continuous devices. Annu. Rev. Phys. Chem., 65: 365–393, 2014. ISSN 0066-426X, 1545-1593. 10.1146/​annurev-physchem-040513-103724.
https:/​/​doi.org/​10.1146/​annurev-physchem-040513-103724

[66] Hyukjoon Kwon, Hyunseok Jeong, David Jennings, Benjamin Yadin, and M S Kim. Clock–Work Trade-Off relation for coherence in quantum thermodynamics. Phys. Rev. Lett., 120 (15): 150602, April 2018. ISSN 0031-9007. 10.1103/​PhysRevLett.120.150602.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.150602

[67] C L Latune, I Sinayskiy, and F Petruccione. Quantum coherence, many-body correlations, and non-thermal effects for autonomous thermal machines. Sci. Rep., 9 (1): 1–13, February 2019. ISSN 2045-2322, 2045-2322. 10.1038/​s41598-019-39300-4.
https:/​/​doi.org/​10.1038/​s41598-019-39300-4

[68] Thao P Le, Jesper Levinsen, Kavan Modi, Meera M Parish, and Felix A Pollock. Spin-chain model of a many-body quantum battery. Phys. Rev. A, 97 (2): 022106, February 2018a. ISSN 1050-2947. 10.1103/​PhysRevA.97.022106.
https:/​/​doi.org/​10.1103/​PhysRevA.97.022106

[69] Thao P Le, Jesper Levinsen, Kavan Modi, Meera M Parish, and Felix A Pollock. Spin-chain model of a many-body quantum battery. Phys. Rev. A, 97 (2): 022106, February 2018b. ISSN 1050-2947. 10.1103/​PhysRevA.97.022106.
https:/​/​doi.org/​10.1103/​PhysRevA.97.022106

[70] Lorenzo Leone, Salvatore F E Oliviero, and Alioscia Hamma. Isospectral twirling and quantum chaos. November 2020. URL http:/​/​arxiv.org/​abs/​2011.06011.
arXiv:2011.06011

[71] Amikam Levy, Robert Alicki, and Ronnie Kosloff. Quantum refrigerators and the third law of thermodynamics. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 85 (6 Pt 1): 061126, June 2012. ISSN 1539-3755, 1550-2376. 10.1103/​PhysRevE.85.061126.
https:/​/​doi.org/​10.1103/​PhysRevE.85.061126

[72] D A Lidar, I L Chuang, and K B Whaley. Decoherence-Free subspaces for quantum computation. Phys. Rev. Lett., 81 (12): 2594–2597, September 1998. ISSN 0031-9007. 10.1103/​PhysRevLett.81.2594.
https:/​/​doi.org/​10.1103/​PhysRevLett.81.2594

[73] Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Ann. Phys., 16 (3): 407–466, December 1961. ISSN 0003-4916. 10.1016/​0003-4916(61)90115-4.
https:/​/​doi.org/​10.1016/​0003-4916(61)90115-4

[74] Noah Linden, Sandu Popescu, and Paul Skrzypczyk. How small can thermal machines be? the smallest possible refrigerator. Phys. Rev. Lett., 105 (13): 130401, September 2010. ISSN 0031-9007. 10.1103/​PhysRevLett.105.130401.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.130401

[75] Matteo Lostaglio, David Jennings, and Terry Rudolph. Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nat. Commun., 6 (1): 1–9, March 2015a. ISSN 2041-1723, 2041-1723. 10.1038/​ncomms7383.
https:/​/​doi.org/​10.1038/​ncomms7383

[76] Matteo Lostaglio, Kamil Korzekwa, David Jennings, and Terry Rudolph. Quantum coherence, Time-Translation symmetry, and thermodynamics. Phys. Rev. X, 5 (2): 021001, April 2015b. 10.1103/​PhysRevX.5.021001.
https:/​/​doi.org/​10.1103/​PhysRevX.5.021001

[77] Matteo Lostaglio, Markus P Müller, and Michele Pastena. Stochastic independence as a resource in Small-Scale thermodynamics. Phys. Rev. Lett., 115 (15): 150402, October 2015c. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.115.150402.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.150402

[78] Iman Marvian, Robert W Spekkens, and Paolo Zanardi. Quantum speed limits, coherence, and asymmetry. Phys. Rev. A, 93 (5): 052331, May 2016. ISSN 1050-2947. 10.1103/​PhysRevA.93.052331.
https:/​/​doi.org/​10.1103/​PhysRevA.93.052331

[79] Lluís Masanes and Jonathan Oppenheim. A general derivation and quantification of the third law of thermodynamics. Nat. Commun., 8 (1): 1–7, March 2017. ISSN 2041-1723, 2041-1723. 10.1038/​ncomms14538.
https:/​/​doi.org/​10.1038/​ncomms14538

[80] Markus P Müller. Correlating thermal machines and the second law at the nanoscale. Phys. Rev. X, 8 (4): 041051, December 2018. 10.1103/​PhysRevX.8.041051.
https:/​/​doi.org/​10.1103/​PhysRevX.8.041051

[81] Salvatore Francesco Emanuele Oliviero, Lorenzo Leone, Francesco Caravelli, and Alioscia Hamma. Random matrix theory of the isospectral twirling. SciPost Phys., 10 (3), March 2021. ISSN 2542-4653. 10.21468/​scipostphys.10.3.076.
https:/​/​doi.org/​10.21468/​scipostphys.10.3.076

[82] G Massimo Palma, Kalle-Antti Suominen, and Artur Ekert. Quantum computers and dissipation. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 452 (1946): 567–584, January 1996. 10.1098/​rspa.1996.0029.
https:/​/​doi.org/​10.1098/​rspa.1996.0029

[83] Martí Perarnau-Llobet and Raam Uzdin. Collective operations can extremely reduce work fluctuations. New J. Phys., 21 (8): 083023, August 2019. ISSN 1367-2630. 10.1088/​1367-2630/​ab36a9.
https:/​/​doi.org/​10.1088/​1367-2630/​ab36a9

[84] F Petruccione and H.-P. Breuer, editors. The Theory of Open Quantum Systems. Oxford University Press, 2002. 10.1093/​acprof:oso/​9780199213900.001.0001.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001

[85] James Q Quach and William J Munro. Using dark states to charge and stabilize open quantum batteries. Phys. Rev. Applied, 14 (2): 024092, August 2020. 10.1103/​PhysRevApplied.14.024092.
https:/​/​doi.org/​10.1103/​PhysRevApplied.14.024092

[86] A T Rezakhani, W-J Kuo, A Hamma, D A Lidar, and P Zanardi. Quantum adiabatic brachistochrone. Phys. Rev. Lett., 103 (8): 080502, August 2009. ISSN 0031-9007. 10.1103/​PhysRevLett.103.080502.
https:/​/​doi.org/​10.1103/​PhysRevLett.103.080502

[87] Jonathan G Richens and Lluis Masanes. Work extraction from quantum systems with bounded fluctuations in work. Nat. Commun., 7 (1): 1–7, November 2016. ISSN 2041-1723, 2041-1723. 10.1038/​ncomms13511.
https:/​/​doi.org/​10.1038/​ncomms13511

[88] Davide Rossini, Gian Marcello Andolina, Dario Rosa, Matteo Carrega, and Marco Polini. Quantum advantage in the charging process of Sachdev-Ye-Kitaev batteries. Phys. Rev. Lett., 125 (23): 236402, December 2020. ISSN 0031-9007. 10.1103/​PhysRevLett.125.236402.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.236402

[89] M. A. Schlosshauer, editor. Decoherence and the quantum-to-classical transition. Springer Verlag, Berlin, 2007. 10.1007/​978-3-540-35775-9.
https:/​/​doi.org/​10.1007/​978-3-540-35775-9

[90] Paul Skrzypczyk, Anthony J Short, and Sandu Popescu. Work extraction and thermodynamics for individual quantum systems. Nat. Commun., 5 (1): 1–8, June 2014. ISSN 2041-1723, 2041-1723. 10.1038/​ncomms5185.
https:/​/​doi.org/​10.1038/​ncomms5185

[91] Carlo Sparaciari, Jonathan Oppenheim, and Tobias Fritz. Resource theory for work and heat. Phys. Rev. A, 96 (5): 052112, November 2017. ISSN 1050-2947. 10.1103/​PhysRevA.96.052112.
https:/​/​doi.org/​10.1103/​PhysRevA.96.052112

[92] Alexander Streltsov, Uttam Singh, Himadri Shekhar Dhar, Manabendra Nath Bera, and Gerardo Adesso. Measuring quantum coherence with entanglement. Phys. Rev. Lett., 115 (2): 020403, July 2015. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.115.020403.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.020403

[93] Alexander Streltsov, Gerardo Adesso, and Martin B Plenio. Colloquium: Quantum coherence as a resource. Rev. Mod. Phys., 89 (4): 041003, October 2017. ISSN 0034-6861. 10.1103/​RevModPhys.89.041003.
https:/​/​doi.org/​10.1103/​RevModPhys.89.041003

[94] Georgios Styliaris, Namit Anand, Lorenzo Campos Venuti, and Paolo Zanardi. Quantum coherence and the localization transition. Phys. Rev. B Condens. Matter, 100 (22): 224204, December 2019. ISSN 0163-1829. 10.1103/​PhysRevB.100.224204.
https:/​/​doi.org/​10.1103/​PhysRevB.100.224204

[95] Gentaro Watanabe, B Prasanna Venkatesh, Peter Talkner, and Adolfo Del Campo. Quantum performance of thermal machines over many cycles. Phys. Rev. Lett., 118 (5): 050601, February 2017. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.118.050601.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.050601

[96] H Wilming, R Gallego, and J Eisert. Second law of thermodynamics under control restrictions. Phys Rev E, 93: 042126, April 2016. ISSN 2470-0053, 2470-0045. 10.1103/​PhysRevE.93.042126.
https:/​/​doi.org/​10.1103/​PhysRevE.93.042126

[97] Yan-Dong Wu and Xu-Qing Liu. A short note on the frobenius norm of the commutator. Math. Notes, 87 (5): 903–907, June 2010. ISSN 0001-4346, 1573-8876. 10.1134/​S0001434610050305.
https:/​/​doi.org/​10.1134/​S0001434610050305

[98] Bin Yan and Nikolai A Sinitsyn. Recovery of damaged information and the Out-of-Time-Ordered correlators. Phys. Rev. Lett., 125 (4): 040605, July 2020. ISSN 0031-9007. 10.1103/​PhysRevLett.125.040605.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.040605

[99] Bin Yan, Lukasz Cincio, and Wojciech H Zurek. Information scrambling and loschmidt echo. Phys. Rev. Lett., 124 (16): 160603, April 2020. ISSN 0031-9007. 10.1103/​PhysRevLett.124.160603.
https:/​/​doi.org/​10.1103/​PhysRevLett.124.160603

[100] P Zanardi and M Rasetti. Noiseless quantum codes. Phys. Rev. Lett., 79 (17): 3306–3309, October 1997. ISSN 0031-9007. 10.1103/​PhysRevLett.79.3306.
https:/​/​doi.org/​10.1103/​PhysRevLett.79.3306

[101] Paolo Zanardi. Entanglement of quantum evolutions. Phys. Rev. A, 63 (4): 040304, March 2001. ISSN 1050-2947. 10.1103/​PhysRevA.63.040304.
https:/​/​doi.org/​10.1103/​PhysRevA.63.040304

[102] Paolo Zanardi and Nikola Paunković. Ground state overlap and quantum phase transitions. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 74 (3 Pt 1): 031123, September 2006. ISSN 1539-3755. 10.1103/​PhysRevE.74.031123.
https:/​/​doi.org/​10.1103/​PhysRevE.74.031123

[103] Paolo Zanardi, Christof Zalka, and Lara Faoro. Entangling power of quantum evolutions. Phys. Rev. A, 62 (3): 030301, August 2000. ISSN 1050-2947. 10.1103/​PhysRevA.62.030301.
https:/​/​doi.org/​10.1103/​PhysRevA.62.030301

[104] J Zhang, G Pagano, P W Hess, A Kyprianidis, P Becker, H Kaplan, A V Gorshkov, Z-X Gong, and C Monroe. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature, 551 (7682): 601–604, November 2017. ISSN 0028-0836. 10.1038/​nature24654.
https:/​/​doi.org/​10.1038/​nature24654

[105] Karol Życzkowski, Paweł Horodecki, Anna Sanpera, and Maciej Lewenstein. Volume of the set of separable states. Phys. Rev. A, 58 (2): 883–892, August 1998. ISSN 1050-2947. 10.1103/​PhysRevA.58.883.
https:/​/​doi.org/​10.1103/​PhysRevA.58.883

Cited by

[1] Anna Delmonte, Alba Crescente, Matteo Carrega, Dario Ferraro, and Maura Sassetti, "Characterization of a Two-Photon Quantum Battery: Initial Conditions, Stability and Work Extraction", Entropy 23 5, 612 (2021).

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