Monotones in General Resource Theories

Tomáš Gonda; Robert W. Spekkens

doi:10.32408/compositionality-5-7

ABSTRACT

A central problem in the study of resource theories is to find functions that are nonincreasing under resource conversions — termed monotones — in order to quantify resourcefulness. Various constructions of monotones appear in many different concrete resource theories. How general are these constructions? What are the necessary conditions on a resource theory for a given construction to be applicable? To answer these questions, we introduce a broad scheme for constructing monotones. It involves finding an order-preserving map from the preorder of resources of interest to a distinct preorder for which nontrivial monotones are previously known or can be more easily constructed; these monotones are then pulled back through the map. In one of the two main classes we study, the preorder of resources is mapped to a preorder of sets of resources, where the order relation is set inclusion, such that monotones can be defined via maximizing or minimizing the value of a function within these sets. In the other class, the preorder of resources is mapped to a preorder of tuples of resources, and one pulls back monotones that measure the amount of distinguishability of the different elements of the tuple (hence its information content). Monotones based on contractions arise naturally in the latter class, and, more surprisingly, so do weight and robustness measures. In addition to capturing many standard monotone constructions, our scheme also suggests significant generalizations of these. In order to properly capture the breadth of applicability of our results, we present them within a novel abstract framework for resource theories in which the notion of composition is independent of the types of the resources involved (i.e., whether they are states, channels, combs, etc.).

► BibTeX data

@article{Gonda2023monotonesingeneral,
  doi = {10.32408/compositionality-5-7},
  url = {https://doi.org/10.32408/compositionality-5-7},
  title = {Monotones in {G}eneral {R}esource {T}heories},
  author = {Gonda, Tom{\'{a}}{\v{s}} and Spekkens, Robert W.},
  journal = {{Compositionality}},
  issn = {2631-4444},
  publisher = {{Compositionality}},
  volume = {5},
  pages = {7},
  month = aug,
  year = {2023}
}

► References

[1] Samson Abramsky, Rui Soares Barbosa, and Shane Mansfield. Contextual fraction as a measure of contextuality. Physical review letters, 119 (5): 050504, 2017. https://doi.org/10.1103/PhysRevLett.119.050504.
https://doi.org/10.1103/PhysRevLett.119.050504

[2] Jonathan Barrett, Adrian Kent, and Stefano Pironio. Maximally nonlocal and monogamous quantum correlations. Physical review letters, 97 (17): 170409, 2006. https://doi.org/10.1103/PhysRevLett.97.170409.
https://doi.org/10.1103/PhysRevLett.97.170409

[3] Fernando Brandão, Michał Horodecki, Jonathan Oppenheim, Joseph M. Renes, and Robert W. Spekkens. Resource theory of quantum states out of thermal equilibrium. Physical review letters, 111 (25): 250404, 2013. https://doi.org/10.1103/PhysRevLett.111.250404,.
https://doi.org/10.1103/PhysRevLett.111.250404

[4] Fernando Brandão, Michał Horodecki, Nelly Ng, Jonathan Oppenheim, and Stephanie Wehner. The second laws of quantum thermodynamics. Proceedings of the National Academy of Sciences, 112 (11): 3275–3279, 2015. https://doi.org/10.1073/pnas.1411728112.
https://doi.org/10.1073/pnas.1411728112

[5] Giulio Chiribella, Giacomo M. D'Ariano, and Paolo Perinotti. Quantum circuit architecture. Phys. Rev. Lett., 101: 060401, Aug 2008. https://doi.org/10.1103/PhysRevLett.101.060401.
https://doi.org/10.1103/PhysRevLett.101.060401

[6] Eric Chitambar and Gilad Gour. Quantum resource theories. Reviews of Modern Physics, 91 (2): 025001, 2019. https://doi.org/10.1103/RevModPhys.91.025001.
https://doi.org/10.1103/RevModPhys.91.025001

[7] Eric Chitambar, Debbie Leung, Laura Mančinska, Maris Ozols, and Andreas Winter. Everything you always wanted to know about locc (but were afraid to ask). Communications in Mathematical Physics, 328 (1): 303–326, May 2014. ISSN 1432-0916. https://doi.org/10.1007/s00220-014-1953-9.
https://doi.org/10.1007/s00220-014-1953-9

[8] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Physical review letters, 23 (15): 880, 1969. https://doi.org/10.1103/PhysRevLett.23.880,.
https://doi.org/10.1103/PhysRevLett.23.880

[9] Bob Coecke and Aleks Kissinger. Picturing quantum processes. In International Conference on Theory and Application of Diagrams, pages 28–31. Springer, 2018. https://doi.org/10.1017/9781316219317.
https://doi.org/10.1017/9781316219317

[10] Bob Coecke and Keye Martin. A partial order on classical and quantum states. In New Structures for Physics, pages 593–683. Springer, 2010. https://doi.org/10.1007/978-3-642-12821-9_10.
https://doi.org/10.1007/978-3-642-12821-9_10

[11] Bob Coecke, Tobias Fritz, and Robert W. Spekkens. A mathematical theory of resources. Information and Computation, 250: 59 – 86, 2016. ISSN 0890-5401. https://doi.org/10.1016/j.ic.2016.02.008.
https://doi.org/10.1016/j.ic.2016.02.008

[12] Geir Dahl. Matrix majorization. Linear Algebra and its Applications, 288: 53 – 73, 1999. ISSN 0024-3795. https://doi.org/10.1016/S0024-3795(98)10175-1.
https://doi.org/10.1016/S0024-3795(98)10175-1

[13] Igor Devetak, Aram W. Harrow, and Andreas J. Winter. A resource framework for quantum shannon theory. IEEE Transactions on Information Theory, 54 (10): 4587–4618, 2008. https://doi.org/10.1109/TIT.2008.928980.
https://doi.org/10.1109/TIT.2008.928980

[14] Andrés F. Ducuara and Paul Skrzypczyk. Weight of informativeness, state exclusion games and excludible information. arXiv:1908.10347, 2019. https://doi.org/10.48550/arXiv.1908.10347.
https://doi.org/10.48550/arXiv.1908.10347
arXiv:1908.10347

[15] Tobias Fritz. Resource convertibility and ordered commutative monoids. Mathematical Structures in Computer Science, 27 (6): 850–938, 2017. https://doi.org/10.1017/S0960129515000444.
https://doi.org/10.1017/S0960129515000444

[16] Tomáš Gonda. Resource Theories as Quantale Modules. PhD thesis, University of Waterloo, 2021. https://doi.org/10.48550/arXiv.2112.02349.
https://doi.org/10.48550/arXiv.2112.02349

[17] Gilad Gour and Marco Tomamichel. Optimal extensions of resource measures and their applications. Physical Review A, 102 (6): 062401, 2020. https://doi.org/10.1103/PhysRevA.102.062401.
https://doi.org/10.1103/PhysRevA.102.062401

[18] Gilad Gour and Andreas Winter. How to quantify a dynamical quantum resource. Physical review letters, 123 (15): 150401, 2019. https://doi.org/10.1103/PhysRevLett.123.150401.
https://doi.org/10.1103/PhysRevLett.123.150401

[19] Gilad Gour, Iman Marvian, and Robert W. Spekkens. Measuring the quality of a quantum reference frame: The relative entropy of frameness. Physical Review A, 80 (1): 012307, 2009. https://doi.org/10.1103/PhysRevA.80.012307.
https://doi.org/10.1103/PhysRevA.80.012307

[20] Gilad Gour, Markus P. Müller, Varun Narasimhachar, Robert W. Spekkens, and Nicole Y. Halpern. The resource theory of informational nonequilibrium in thermodynamics. Physics Reports, 583: 1 – 58, 2015. ISSN 0370-1573. https://doi.org/10.1016/j.physrep.2015.04.003.
https://doi.org/10.1016/j.physrep.2015.04.003

[21] George Grätzer. Universal algebra. Springer Science & Business Media, 2008. https://doi.org/10.1007/978-0-387-77487-9.
https://doi.org/10.1007/978-0-387-77487-9

[22] Gus Gutoski and John Watrous. Toward a general theory of quantum games. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 565–574. ACM, 2007. https://doi.org/10.1145/1250790.1250873.
https://doi.org/10.1145/1250790.1250873

[23] Michał Horodecki and Jonathan Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nature communications, 4: 2059, 2013. https://doi.org/10.1038/ncomms3059.
https://doi.org/10.1038/ncomms3059

[24] Michał Horodecki, Paweł Horodecki, and Jonathan Oppenheim. Reversible transformations from pure to mixed states and the unique measure of information. Physical Review A, 67 (6): 062104, 2003. https://doi.org/10.1103/PhysRevA.67.062104.
https://doi.org/10.1103/PhysRevA.67.062104

[25] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Reviews of Modern Physics, 81: 865–942, April 2009. https://doi.org/10.1103/RevModPhys.81.865.
https://doi.org/10.1103/RevModPhys.81.865

[26] Dominik Janzing, Pawel Wocjan, Robert Zeier, Rubino Geiss, and Thomas Beth. Thermodynamic cost of reliability and low temperatures: tightening landauer's principle and the second law. International Journal of Theoretical Physics, 39 (12): 2717–2753, 2000. https://doi.org/10.1023/A:1026422630734.
https://doi.org/10.1023/A:1026422630734

[27] Nathaniel Johnston, Chi-Kwong Li, Sarah Plosker, Yiu-Tung Poon, and Bartosz Regula. Evaluating the robustness of k-coherence and k-entanglement. Physical Review A, 98 (2): 022328, 2018. https://doi.org/10.1103/PhysRevA.98.022328.
https://doi.org/10.1103/PhysRevA.98.022328

[28] Nathan Killoran, Frank E. S. Steinhoff, and Martin B. Plenio. Converting nonclassicality into entanglement. Physical review letters, 116 (8): 080402, 2016. https://doi.org/10.1103/PhysRevLett.116.080402.
https://doi.org/10.1103/PhysRevLett.116.080402

[29] Lea Kraemer and Lídia Del Rio. Currencies in resource theories. Entropy, 23 (6): 755, 2021. https://doi.org/10.3390/e23060755.
https://doi.org/10.3390/e23060755

[30] Felix Leditzky, Eneet Kaur, Nilanjana Datta, and Mark M. Wilde. Approaches for approximate additivity of the holevo information of quantum channels. Phys. Rev. A, 97: 012332, Jan 2018. https://doi.org/10.1103/PhysRevA.97.012332.
https://doi.org/10.1103/PhysRevA.97.012332

[31] Elliott H. Lieb and Jakob Yngvason. The physics and mathematics of the second law of thermodynamics. Physics Reports, 310 (1): 1–96, 1999. https://doi.org/10.1016/S0370-1573(98)00082-9.
https://doi.org/10.1016/S0370-1573(98)00082-9

[32] Elliott H. Lieb and Jakob Yngvason. The entropy concept for non-equilibrium states. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469 (2158): 20130408, 2013. https://doi.org/10.1098/rspa.2013.0408.
https://doi.org/10.1098/rspa.2013.0408

[33] Yunchao Liu and Xiao Yuan. Operational resource theory of quantum channels. Physical Review Research, 2 (1): 012035, 2020. https://doi.org/10.1103/PhysRevResearch.2.012035.
https://doi.org/10.1103/PhysRevResearch.2.012035

[34] Zi-Wen Liu and Andreas Winter. Resource theories of quantum channels and the universal role of resource erasure. arXiv:1904.04201, 2019. https://doi.org/10.48550/arXiv.1904.04201.
https://doi.org/10.48550/arXiv.1904.04201
arXiv:1904.04201

[35] Albert W. Marshall, Ingram Olkin, and Barry C. Arnold. Inequalities: Theory of Majorization and Its Applications. Springer Series in Statistics. Springer, New York, NY, second edition edition, 2011. https://doi.org/10.1007/978-0-387-68276-1.
https://doi.org/10.1007/978-0-387-68276-1

[36] Iman Marvian. Symmetry, asymmetry and quantum information. PhD thesis, University of Waterloo, 2012. URL http://hdl.handle.net/10012/7088.
http://hdl.handle.net/10012/7088

[37] Iman Marvian and Robert W. Spekkens. The theory of manipulations of pure state asymmetry: I. basic tools, equivalence classes and single copy transformations. New Journal of Physics, 15 (3): 033001, 2013. https://doi.org/10.1088/1367-2630/15/3/033001.
https://doi.org/10.1088/1367-2630/15/3/033001

[38] Iman Marvian and Robert W. Spekkens. Extending noether’s theorem by quantifying the asymmetry of quantum states. Nature communications, 5: 3821, 2014. https://doi.org/10.1038/ncomms4821.
https://doi.org/10.1038/ncomms4821

[39] Michael A. Nielsen. Conditions for a class of entanglement transformations. Physical Review Letters, 83 (2): 436, 1999. https://doi.org/10.1103/PhysRevLett.83.436.
https://doi.org/10.1103/PhysRevLett.83.436

[40] Dénes Petz. Quasi-entropies for finite quantum systems. Reports on Mathematical Physics, 23 (1): 57–65, 1986. ISSN 0034-4877. https://doi.org/10.1016/0034-4877(86)90067-4.
https://doi.org/10.1016/0034-4877(86)90067-4

[41] Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics, 24 (3): 379–385, 1994. https://doi.org/10.1007/BF02058098.
https://doi.org/10.1007/BF02058098

[42] Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel. Measurement-based quantum computation on cluster states. Physical review A, 68 (2): 022312, 2003. https://doi.org/10.1103/PhysRevA.68.022312.
https://doi.org/10.1103/PhysRevA.68.022312

[43] Bartosz Regula. Convex geometry of quantum resource quantification. Journal of Physics A: Mathematical and Theoretical, 51 (4): 045303, Dec 2017. https://doi.org/10.1088/1751-8121/aa9100.
https://doi.org/10.1088/1751-8121/aa9100

[44] Bartosz Regula, Marco Piani, Marco Cianciaruso, Thomas R. Bromley, Alexander Streltsov, and Gerardo Adesso. Converting multilevel nonclassicality into genuine multipartite entanglement. New Journal of Physics, 20 (3): 033012, 2018. https://doi.org/10.1088/1367-2630/aaae9d.
https://doi.org/10.1088/1367-2630/aaae9d

[45] Kimmo I. Rosenthal. Quantales and their applications, volume 234. Longman Scientific and Technical, 1990.

[46] David Schmid, Thomas C. Fraser, Ravi Kunjwal, Ana Belen Sainz, Elie Wolfe, and Robert W. Spekkens. Understanding the interplay of entanglement and nonlocality: motivating and developing a new branch of entanglement theory. arXiv:2004.09194, 2020a. https://doi.org/10.48550/arXiv.2004.09194.
https://doi.org/10.48550/arXiv.2004.09194
arXiv:2004.09194

[47] David Schmid, Denis Rosset, and Francesco Buscemi. The type-independent resource theory of local operations and shared randomness. Quantum, 4: 262, 2020b. https://doi.org/10.22331/q-2020-04-30-262.
https://doi.org/10.22331/q-2020-04-30-262

[48] Claude E. Shannon. A mathematical theory of communication. Bell system technical journal, 27 (3): 379–423, 1948. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.
https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

[49] Paul Skrzypczyk and Noah Linden. Robustness of measurement, discrimination games, and accessible information. Physical review letters, 122 (14): 140403, 2019. https://doi.org/10.1103/PhysRevLett.122.140403.
https://doi.org/10.1103/PhysRevLett.122.140403

[50] Carlo Sparaciari, Jonathan Oppenheim, and Tobias Fritz. Resource theory for work and heat. Phys. Rev. A, 96: 052112, Nov 2017. https://doi.org/10.1103/PhysRevA.96.052112.
https://doi.org/10.1103/PhysRevA.96.052112

[51] Carlo Sparaciari, Lídia Del Rio, Carlo Maria Scandolo, Philippe Faist, and Jonathan Oppenheim. The first law of general quantum resource theories. Quantum, 4: 259, 2020. https://doi.org/10.22331/q-2020-04-30-259.
https://doi.org/10.22331/q-2020-04-30-259

[52] J. Sperling and W. Vogel. Convex ordering and quantification of quantumness. Physica Scripta, 90 (7): 074024, 2015. https://doi.org/10.1088/0031-8949/90/7/074024.
https://doi.org/10.1088/0031-8949/90/7/074024

[53] Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio. Colloquium: Quantum coherence as a resource. Reviews of Modern Physics, 89 (4): 041003, 2017. https://doi.org/10.1103/RevModPhys.89.041003,.
https://doi.org/10.1103/RevModPhys.89.041003

[54] Leó Szilard. Über die entropieverminderung in einem thermodynamischen system bei eingriffen intelligenter wesen. Zeitschrift für Physik, 53 (11): 840–856, Nov 1929. ISSN 0044-3328. https://doi.org/10.1007/BF01341281.
https://doi.org/10.1007/BF01341281

[55] Ryuji Takagi and Bartosz Regula. General resource theories in quantum mechanics and beyond: operational characterization via discrimination tasks. Physical Review X, 9 (3): 031053, 2019. https://doi.org/10.1103/PhysRevX.9.031053,.
https://doi.org/10.1103/PhysRevX.9.031053

[56] Ryuji Takagi, Bartosz Regula, Kaifeng Bu, Zi-Wen Liu, and Gerardo Adesso. Operational advantage of quantum resources in subchannel discrimination. Physical review letters, 122 (14): 140402, 2019. https://doi.org/10.1103/PhysRevLett.122.140402.
https://doi.org/10.1103/PhysRevLett.122.140402

[57] Barbara M. Terhal and Paweł Horodecki. Schmidt number for density matrices. Phys. Rev. A, 61: 040301, Mar 2000. https://doi.org/10.1103/PhysRevA.61.040301.
https://doi.org/10.1103/PhysRevA.61.040301

[58] Roope Uola, Tristan Kraft, Jiangwei Shang, Xiao-Dong Yu, and Otfried Gühne. Quantifying quantum resources with conic programming. Physical review letters, 122 (13): 130404, 2019. https://doi.org/10.1103/PhysRevLett.122.130404.
https://doi.org/10.1103/PhysRevLett.122.130404

[59] Joan A. Vaccaro, Fabio Anselmi, Howard M. Wiseman, and Kurt Jacobs. Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules. Physical Review A, 77 (3): 032114, 2008. https://doi.org/10.1103/PhysRevA.77.032114.
https://doi.org/10.1103/PhysRevA.77.032114

[60] Vlatko Vedral, Martin B. Plenio, Michael A. Rippin, and Peter L. Knight. Quantifying entanglement. Physical Review Letters, 78 (12): 2275, 1997. https://doi.org/10.1103/PhysRevLett.78.2275.
https://doi.org/10.1103/PhysRevLett.78.2275

[61] Arthur F. Veinott Jr. Least d-majorized network flows with inventory and statistical applications. Management Science, 17 (9): 547–567, 1971. https://doi.org/10.1287/mnsc.17.9.547.
https://doi.org/10.1287/mnsc.17.9.547

[62] Xin Wang and Mark M. Wilde. Resource theory of asymmetric distinguishability. Physical Review Research, 1 (3): 033170, 2019. https://doi.org/10.1103/PhysRevResearch.1.033170.
https://doi.org/10.1103/PhysRevResearch.1.033170

[63] John Watrous. The theory of quantum information. Cambridge University Press, 2018. https://doi.org/10.1017/9781316848142.
https://doi.org/10.1017/9781316848142

[64] Mirjam Weilenmann, Lea Kraemer, Philippe Faist, and Renato Renner. Axiomatic relation between thermodynamic and information-theoretic entropies. Physical review letters, 117 (26): 260601, 2016. https://doi.org/10.1103/PhysRevLett.117.260601.
https://doi.org/10.1103/PhysRevLett.117.260601

[65] Matt Wilson and Giulio Chiribella. A mathematical framework for transformations of physical processes. arXiv:2204.04319, 2022. https://doi.org/10.48550/arXiv.2204.04319.
https://doi.org/10.48550/arXiv.2204.04319
arXiv:2204.04319

[66] Elie Wolfe, David Schmid, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens. Quantifying bell: The resource theory of nonclassicality of common-cause boxes. Quantum, 4: 280, 2020. https://doi.org/10.22331/q-2020-06-08-280.
https://doi.org/10.22331/q-2020-06-08-280

Cited by

Could not fetch Crossref cited-by data during last attempt 2024-11-05 05:31:41: Could not fetch cited-by data for 10.32408/compositionality-5-7 from Crossref. This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2024-11-05 05:31:41).

This Paper is published in Compositionality under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

Published:	2023-08-09, volume 5, page 7
Eprint:	arXiv:1912.07085v3
Doi:	https://doi.org/10.32408/compositionality-5-7
Citation:	Compositionality 5, 7 (2023).

Compositionality

The open-access journal for the mathematics of composition

ABSTRACT

► BibTeX data

► References

Cited by