The geometry of passivity for quantum systems and a novel elementary derivation of the Gibbs state

Nikolaos Koukoulekidis; Rhea Alexander; Thomas Hebdige; David Jennings

doi:10.22331/q-2021-03-15-411

The geometry of passivity for quantum systems and a novel elementary derivation of the Gibbs state

Nikolaos Koukoulekidis¹, Rhea Alexander¹, Thomas Hebdige^1,2, and David Jennings^1,3,4

¹Department of Physics, Imperial College London, London SW7 2AZ, UK
²Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK
³School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK
⁴Department of Physics, University of Oxford, Oxford, OX1 3PU, UK

Published:	2021-03-15, volume 5, page 411
Eprint:	arXiv:1912.07968v3
Doi:	https://doi.org/10.22331/q-2021-03-15-411
Citation:	Quantum 5, 411 (2021).

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Abstract

Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we reformulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it also provides novel analysis for non-equilibrium quantum systems. We show that, to every passive state, one can associate a simple convex shape in a $2$-dimensional plane, and that the area of this shape measures the degree to which the system deviates from the manifold of equilibrium states. This provides a novel geometric measure of athermality with relations to both ergotropy and $\beta$--athermality.

Popular summary

When can useful energy be extracted from a physical system? If no mechanical energy can be extracted in a reversible manner, the system is said to be in a state of 'passivity' – a basic notion of thermodynamics. However, with multiple copies this passivity eventually breaks unless the state is exactly in an equilibrium (Gibbs) state. Here we introduce a geometric framework to describe the structure of passive states via a surprisingly simple geometric figure in the 2-d plane. This representation makes the derivation of the equilibrium Gibbs state almost trivial and shows that non-equilibrium properties in the thermodynamic limit are in one-to-one correspondence with non-trivial polygons in the planar representation. Notably this description is independent of any reference equilibrium temperature. Besides being of conceptual interest, this approach may also find application, for example, in analysing what properties of finite-sized quantum systems are energetically useful.

► BibTeX data

@article{Koukoulekidis2021geometryofpassivity,
  doi = {10.22331/q-2021-03-15-411},
  url = {https://doi.org/10.22331/q-2021-03-15-411},
  title = {The geometry of passivity for quantum systems and a novel elementary derivation of the {G}ibbs state},
  author = {Koukoulekidis, Nikolaos and Alexander, Rhea and Hebdige, Thomas and Jennings, David},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {411},
  month = mar,
  year = {2021}
}

► References

[1] O. Penrose, Foundations of statistical mechanics: a deductive treatment, International series of monographs in natural philosophy (Pergamon Press, 1969).

[2] E. T. Jaynes, Phys. Rev. 106, 620 (1957a).
https://doi.org/10.1103/PhysRev.106.620

[3] E. T. Jaynes, Phys. Rev. 108, 171 (1957b).
https://doi.org/10.1103/PhysRev.108.171

[4] C. Gogolin and J. Eisert, Rep. Prog. Phys 79, 056001 (2016).
https://doi.org/10.1088/0034-4885/79/5/056001

[5] M. A. Cazalilla and M. Rigol, New J. Phys. 12, 055006 (2010).
https://doi.org/10.1088/1367-2630/12/5/055006

[6] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011).
https://doi.org/10.1103/RevModPhys.83.863

[7] W. Pusz and S. L. Woronowicz, Comm. Math. Phys. 58, 273 (1978).
https://doi.org/10.1007/BF01614224

[8] A. Lenard, J. Stat. Phys 19, 575 (1978).
https://doi.org/10.1007/BF01011769

[9] P. Skrzypczyk, R. Silva, and N. Brunner, Phys. Rev. E 91, 052133 (2015).
https://doi.org/10.1103/PhysRevE.91.052133

[10] O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics: Equilibrium States. Models in Quantum Statistical Mechanics (Springer, 2003).
https://doi.org/10.1007/978-3-662-09089-3

[11] M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, J. Tura, and A. Acín, Phys. Rev. E 92, 042147 (2015).
https://doi.org/10.1103/PhysRevE.92.042147

[12] C. Sparaciari, D. Jennings, and J. Oppenheim, Nat. Commun 8 (2017a), 10.1038/s41467-017-01505-4.
https://doi.org/10.1038/s41467-017-01505-4

[13] R. Salvia and V. Giovannetti, Quantum 4, 274 (2020).
https://doi.org/10.22331/q-2020-05-28-274

[14] A. M. Alhambra, G. Styliaris, N. A. Rodríguez-Briones, J. Sikora, and E. Martín-Martínez, Phys. Rev. Lett. 123, 190601 (2019).
https://doi.org/10.1103/PhysRevLett.123.190601

[15] M. Alimuddin, T. Guha, and P. Parashar, Phys. Rev. E 102, 022106 (2020).
https://doi.org/10.1103/PhysRevE.102.022106

[16] R. Balian and N. Balazs, Ann. Phys. 179, 97 (1987).
https://doi.org/10.1016/S0003-4916(87)80006-4

[17] M. Lostaglio, D. Jennings, and T. Rudolph, New J. Phys. 19, 043008 (2017).
https://doi.org/10.1088/1367-2630/aa617f

[18] N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Winter, Nat. Commun 7, 12051 (2016).
https://doi.org/10.1038/ncomms12051

[19] Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Nat. Commun 7, 12049 (2016).
https://doi.org/10.1038/ncomms12049

[20] P. Boes, H. Wilming, J. Eisert, and R. Gallego, Nat. Commun 9, 1022 (2018).
https://doi.org/10.1038/s41467-018-03230-y

[21] M. N. Bera, A. Riera, M. Lewenstein, Z. B. Khanian, and A. Winter, Quantum 3, 121 (2019).
https://doi.org/10.22331/q-2019-02-14-121

[22] R. Uzdin and S. Rahav, Phys. Rev. X 8, 021064 (2018).
https://doi.org/10.1103/PhysRevX.8.021064

[23] R. Uzdin and S. Rahav, arXiv:1912.07922 (2019).
https://doi.org/10.1103/PRXQuantum.2.010336
arXiv:1912.07922

[24] E. G. Brown, N. Friis, and M. Huber, New J. Phys. 18, 113028 (2016).
https://doi.org/10.1088/1367-2630/18/11/113028

[25] F. Binder, L. Correa, C. Gogolin, J. Anders, and G. Adesso, Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, Fundamental Theories of Physics (Springer, 2019).
https://doi.org/10.1007/978-3-319-99046-0

[26] A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, EPL 67, 565 (2004).
https://doi.org/10.1209/epl/i2004-10101-2

[27] C. Sparaciari, J. Oppenheim, and T. Fritz, Phys. Rev. A 96, 052112 (2017b).
https://doi.org/10.1103/PhysRevA.96.052112

[28] M. Planck and A. Ogg, Treatise on Thermodynamics (Dover Publications, 1945).

[29] H. Callen, Thermodynamics and an Introduction to Thermostatistics (Wiley, 1985).

[30] K. Riley, M. Hobson, and S. Bence, Am. J. Phys. 67 (1999), 10.2277/0521861535.
https://doi.org/10.2277/0521861535

[31] F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Phys. Rev. Lett. 111, 250404 (2013).
https://doi.org/10.1103/PhysRevLett.111.250404

[32] R. Alicki and M. Fannes, Phys. Rev. E 87, 042123 (2013).
https://doi.org/10.1103/PhysRevE.87.042123

[33] L. Mirsky, Math. Nachr. 20, 171 (1959).
https://doi.org/10.1002/mana.19590200306

[34] G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities (Cambridge University Press, 1988).

[35] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications (Springer, 2011).
https://doi.org/10.1007/978-0-387-68276-1

[36] O. Perron, Math. Ann. 64, 248 (1907).
https://doi.org/10.1007/BF01449896

[37] D. Markham, J. A. Miszczak, Z. Puchała, and K. Życzkowski, Phys. Rev. A 77, 042111 (2008).
https://doi.org/10.1103/PhysRevA.77.042111

[38] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications (Cambridge University Press, 2013).
https://doi.org/10.1017/CBO9781139003858

[39] T. Tao and V. H. Vu, Additive combinatorics., Vol. 105 (Cambridge University Press, 2006).
https://doi.org/10.1017/CBO9780511755149

[40] W. D. Blizard, Notre Dame J. Formal Logic 30, 36 (1988).
https://doi.org/10.1305/ndjfl/1093634995

[41] R. Haag, D. Kastler, and E. B. Trych-Pohlmeyer, Commun. Math. Phys 38, 173 (1974).
https://doi.org/10.1007/BF01651541

Cited by

[1] Uttam Singh, Siddhartha Das, and Nicolas J. Cerf, "Partial order on passive states and Hoffman majorization in quantum thermodynamics", Physical Review Research 3 3, 033091 (2021).

[2] Mir Alimuddin, Tamal Guha, and Preeti Parashar, "Structure of passive states and its implication in charging quantum batteries", Physical Review E 102 2, 022106 (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2021-08-04 14:36:07) and SAO/NASA ADS (last updated successfully 2021-08-04 14:36:08). The list may be incomplete as not all publishers provide suitable and complete citation data.

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