The first law of general quantum resource theories

Carlo Sparaciari1, Lídia del Rio2, Carlo Maria Scandolo3, Philippe Faist4,5, and Jonathan Oppenheim1

1Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom
2Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
3Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK
4Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
5Institute for Quantum Information and Matter, Caltech, Pasadena CA, 91125 USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We extend the tools of quantum resource theories to scenarios in which multiple quantities (or resources) are present, and their interplay governs the evolution of physical systems. We derive conditions for the interconversion of these resources, which generalise the first law of thermodynamics. We study reversibility conditions for multi-resource theories, and find that the relative entropy distances from the invariant sets of the theory play a fundamental role in the quantification of the resources. The first law for general multi-resource theories is a single relation which links the change in the properties of the system during a state transformation and the weighted sum of the resources exchanged. In fact, this law can be seen as relating the change in the relative entropy from different sets of states. In contrast to typical single-resource theories, the notion of free states and invariant sets of states become distinct in light of multiple constraints. Additionally, generalisations of the Helmholtz free energy, and of adiabatic and isothermal transformations, emerge. We thus have a set of laws for general quantum resource theories, which generalise the laws of thermodynamics. We first test this approach on thermodynamics with multiple conservation laws, and then apply it to the theory of local operations under energetic restrictions.

► BibTeX data

► References

[1] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54 (5): 3824, 1996. 10.1103/​PhysRevA.54.3824.

[2] Eric M. Rains. Entanglement purification via separable superoperators. arXiv:quant-ph/​9707002, 1997. URL https:/​/​​abs/​quant-ph/​9707002.

[3] Vlatko Vedral and Martin B. Plenio. Entanglement measures and purification procedures. Physical Review A, 57 (3): 1619–1633, 1998. 10.1103/​PhysRevA.57.1619.

[4] Eric M. Rains. Bound on distillable entanglement. Physical Review A, 60 (1): 179–184, 1999. 10.1103/​PhysRevA.60.179.

[5] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Reviews of Modern Physics, 81 (2): 865–942, 2009. 10.1103/​RevModPhys.81.865.

[6] 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. 10.1023/​A:1026422630734.

[7] 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. 10.1103/​PhysRevA.67.062104.

[8] Lídia del Rio, Johan Åberg, Renato Renner, Oscar Dahlsten, and Vlatko Vedral. The thermodynamic meaning of negative entropy. Nature, 474 (7349): 61–63, 2011. 10.1038/​nature10123.

[9] Oscar C. O. Dahlsten, Renato Renner, Elisabeth Rieper, and Vlatko Vedral. Inadequacy of von Neumann entropy for characterizing extractable work. New Journal of Physics, 13 (053015): 053015, 2011. 10.1088/​1367-2630/​13/​5/​053015.

[10] 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. Physical Review Letters, 111 (25): 250404, 2013. 10.1103/​PhysRevLett.111.250404.

[11] Michał Horodecki and Jonathan Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nature Communications, 4: 2059, 2013. 10.1038/​ncomms3059.

[12] Paul Skrzypczyk, Anthony J. Short, and Sandu Popescu. Work extraction and thermodynamics for individual quantum systems. Nature Communications, 5: 4185, 2014. 10.1038/​ncomms5185.

[13] Rodrigo Gallego, Jens Eisert, and Henrik Wilming. Thermodynamic work from operational principles. New Journal of Physics, 18 (10): 103017, 2016. 10.1088/​1367-2630/​18/​10/​103017.

[14] Gilad Gour and Robert W Spekkens. The resource theory of quantum reference frames: manipulations and monotones. New Journal of Physics, 10 (3): 033023, 2008. 10.1088/​1367-2630/​10/​3/​033023.

[15] 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. 10.1103/​PhysRevA.80.012307.

[16] 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. 10.1088/​1367-2630/​15/​3/​033001.

[17] Andrea Mari and Jens Eisert. Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient. Physical Review Letters, 109 (23): 230503, 2012. 10.1103/​PhysRevLett.109.230503.

[18] Victor Veitch, S. A. Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. The resource theory of stabilizer quantum computation. New Journal of Physics, 16 (1): 013009, 2014. 10.1088/​1367-2630/​16/​1/​013009.

[19] Victor Veitch, Christopher Ferrie, David Gross, and Joseph Emerson. Negative quasi-probability as a resource for quantum computation. New Journal of Physics, 14 (11): 113011, 2012. 10.1088/​1367-2630/​14/​11/​113011.

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

[21] Elliott H. Lieb and Jakob Yngvason. The entropy concept for non-equilibrium states. Proc. R. Soc. A, 469 (2158): 20130408, 2013. 10.1098/​rspa.2013.0408.

[22] Mirjam Weilenmann, Lea Kraemer, Philippe Faist, and Renato Renner. Axiomatic Relation between Thermodynamic and Information-Theoretic Entropies. Physical Review Letters, 117 (26): 260601, 2016. 10.1103/​PhysRevLett.117.260601.

[23] Tobias Fritz. Resource convertibility and ordered commutative monoids. Mathematical Structures in Computer Science, 27: 850–918, 2017. 10.1017/​S0960129515000444.

[24] Lídia del Rio, Lea Kraemer, and Renato Renner. Resource theories of knowledge. arXiv:1511.08818 [cond-mat, physics:math-ph, physics:quant-ph], 2015. URL http:/​/​​abs/​1511.08818.

[25] Bob Coecke, Tobias Fritz, and Robert W. Spekkens. A mathematical theory of resources. Information and Computation, 250: 59–86, 2016. 10.1016/​j.ic.2016.02.008.

[26] Anurag Anshu, Min-Hsiu Hsieh, and Rahul Jain. Quantifying Resources in General Resource Theory with Catalysts. Physical Review Letters, 121 (19): 190504, 2018. 10.1103/​PhysRevLett.121.190504.

[27] 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. 10.1073/​pnas.1411728112.

[28] Sandu Popescu and Daniel Rohrlich. Thermodynamics and the measure of entanglement. Physical Review A, 56 (5): R3319–R3321, 1997. 10.1103/​PhysRevA.56.R3319.

[29] Michał Horodecki, Jonathan Oppenheim, and Ryszard Horodecki. Are the Laws of Entanglement Theory Thermodynamical? Physical Review Letters, 89 (24): 240403, 2002. 10.1103/​PhysRevLett.89.240403.

[30] Fernando G. S. L. Brandão and Martin B. Plenio. Entanglement theory and the second law of thermodynamics. Nature Physics, 4 (11): 873–877, 2008. 10.1038/​nphys1100.

[31] Fernando G. S. L. Brandão and Martin B. Plenio. A Reversible Theory of Entanglement and its Relation to the Second Law. Communications in Mathematical Physics, 295 (3): 829–851, 2010a. 10.1007/​s00220-010-1003-1.

[32] Michał Horodecki and Jonathan Oppenheim. (quantumness in the context of) resource theories. International Journal of Modern Physics B, 27 (01n03): 1345019, 2012. 10.1142/​S0217979213450197.

[33] Fernando G. S. L. Brandão and Gilad Gour. Reversible Framework for Quantum Resource Theories. Physical Review Letters, 115 (7): 070503, 2015. 10.1103/​PhysRevLett.115.070503.

[34] Lluís Masanes and Jonathan Oppenheim. A general derivation and quantification of the third law of thermodynamics. Nature Communications, 8: 14538, 2017. 10.1038/​ncomms14538.

[35] Carlo Sparaciari, Jonathan Oppenheim, and Tobias Fritz. Resource theory for work and heat. Physical Review A, 96 (5): 052112, 2017. 10.1103/​PhysRevA.96.052112.

[36] Manabendra N. Bera, Arnau Riera, Maciej Lewenstein, and Andreas Winter. Thermodynamics as a Consequence of Information Conservation. Quantum, 3: 121, 2019. 10.22331/​q-2019-02-14-121.

[37] Gilad Gour, Markus P. Müller, Varun Narasimhachar, Robert W. Spekkens, and Nicole Yunger Halpern. The resource theory of informational nonequilibrium in thermodynamics. Physics Reports, 583: 1–58, 2015. 10.1016/​j.physrep.2015.04.003.

[38] Johan Åberg. Quantifying Superposition. arXiv:quant-ph/​0612146, 2006. URL http:/​/​​abs/​quant-ph/​0612146.

[39] Tillmann Baumgratz, Marcus Cramer, and Martin B. Plenio. Quantifying Coherence. Physical Review Letters, 113 (14): 140401, 2014. 10.1103/​PhysRevLett.113.140401.

[40] Andreas Winter and Dong Yang. Operational Resource Theory of Coherence. Physical Review Letters, 116 (12): 120404, 2016. 10.1103/​PhysRevLett.116.120404.

[41] David Slepian and Jack K. Wolf. Noiseless coding of correlated information sources. IEEE Transactions on Information Theory, 19 (4): 471–480, 1973. 10.1109/​TIT.1973.1055037.

[42] Michał Horodecki, Jonathan Oppenheim, and Andreas Winter. Partial quantum information. Nature, 436 (7051): 673–676, 2005. 10.1038/​nature03909.

[43] Mehdi Ahmadi, David Jennings, and Terry Rudolph. The Wigner–Araki–Yanase theorem and the quantum resource theory of asymmetry. New Journal of Physics, 15 (1): 013057, 2013. 10.1088/​1367-2630/​15/​1/​013057.

[44] Uttam Singh, Manabendra N. Bera, Himadri S. Dhar, and Arun K. Pati. Maximally coherent mixed states: Complementarity between maximal coherence and mixedness. Physical Review A, 91 (5): 052115, 2015. 10.1103/​PhysRevA.91.052115.

[45] Alexander Streltsov, Eric Chitambar, Swapan Rana, Manabendra N. Bera, Andreas Winter, and Maciej Lewenstein. Entanglement and Coherence in Quantum State Merging. Physical Review Letters, 116 (24): 240405, 2016. 10.1103/​PhysRevLett.116.240405.

[46] Eric Chitambar and Min-Hsiu Hsieh. Relating the Resource Theories of Entanglement and Quantum Coherence. Physical Review Letters, 117 (2): 020402, 2016. 10.1103/​PhysRevLett.117.020402.

[47] Paul Erker, Mark T. Mitchison, Ralph Silva, Mischa P. Woods, Nicolas Brunner, and Marcus Huber. Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time? Physical Review X, 7 (3): 031022, 2017. 10.1103/​PhysRevX.7.031022.

[48] Yelena Guryanova, Sandu Popescu, Anthony J. Short, Ralph Silva, and Paul Skrzypczyk. Thermodynamics of quantum systems with multiple conserved quantities. Nature Communications, 7: 12049, 2016. 10.1038/​ncomms12049.

[49] Nicole Yunger Halpern, Philippe Faist, Jonathan Oppenheim, and Andreas Winter. Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges. Nature Communications, 7: 12051, 2016. 10.1038/​ncomms12051.

[50] Nicole Yunger Halpern and Joseph M. Renes. Beyond heat baths: Generalized resource theories for small-scale thermodynamics. Physical Review E, 93 (2): 022126, 2016. 10.1103/​PhysRevE.93.022126.

[51] Matteo Lostaglio, David Jennings, and Terry Rudolph. Thermodynamic resource theories, non-commutativity and maximum entropy principles. New Journal of Physics, 19 (4): 043008, 2017. 10.1088/​1367-2630/​aa617f.

[52] Nicole Yunger Halpern. Beyond heat baths II: framework for generalized thermodynamic resource theories. Journal of Physics A: Mathematical and Theoretical, 51 (9): 094001, 2018. 10.1088/​1751-8121/​aaa62f.

[53] Sandu Popescu, Ana Belén Sainz, Anthony J. Short, and Andreas Winter. Quantum reference frames and their applications to thermodynamics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2123): 20180111, 2018. 10.1098/​rsta.2018.0111.

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

[55] Joseph M. Renes. Work cost of thermal operations in quantum thermodynamics. The European Physical Journal Plus, 129 (7): 153, 2014. 10.1140/​epjp/​i2014-14153-8.

[56] Charles H. Bennett, David P. DiVincenzo, Christopher A. Fuchs, Tal Mor, Eric Rains, Peter W. Shor, John A. Smolin, and William K. Wootters. Quantum nonlocality without entanglement. Physical Review A, 59 (2): 1070, 1999a. 10.1103/​PhysRevA.59.1070.

[57] Göran Lindblad. Completely positive maps and entropy inequalities. Communications in Mathematical Physics, 40 (2): 147–151, 1975. 10.1007/​BF01609396.

[58] Charles H. Bennett, David P. DiVincenzo, Tal Mor, Peter W. Shor, John A. Smolin, and Barbara M. Terhal. Unextendible Product Bases and Bound Entanglement. Physical Review Letters, 82 (26): 5385–5388, 1999b. 10.1103/​PhysRevLett.82.5385.

[59] Lidia del Rio, Philippe Faist, Jonathan Oppenheim, Carlo Maria Scandolo, and Carlo Sparaciari. In preparation.

[60] Lea Kraemer and Lidia del Rio. Currencies in resource theories. arXiv:1605.01064 [cond-mat, physics:math-ph, physics:quant-ph], 2016. URL https:/​/​​abs/​1605.01064.

[61] K. G. H. Vollbrecht and R. F. Werner. Entanglement measures under symmetry. Physical Review A, 64 (6): 062307, 2001. 10.1103/​PhysRevA.64.062307.

[62] Michał Horodecki. Entanglement Measures. Quantum Information and Computation, 1 (1): 3 – 26, 2001. 10.26421/​QIC1.1.

[63] Fernando G. S. L. Brandão and Martin B. Plenio. A Generalization of Quantum Stein’s Lemma. Communications in Mathematical Physics, 295 (3): 791–828, 2010b. 10.1007/​s00220-010-1005-z.

[64] Barbara Synak-Radtke and Michał Horodecki. On asymptotic continuity of functions of quantum states. Journal of Physics A: Mathematical and General, 39 (26): L423, 2006. 10.1088/​0305-4470/​39/​26/​L02.

[65] Rolf Landauer. Irreversibility and heat generation in the computing process. IBM journal of research and development, 5 (3): 183–191, 1961. 10.1147/​rd.53.0183.

[66] Charles H. Bennett. The thermodynamics of computation—a review. International Journal of Theoretical Physics, 21 (12): 905–940, 1982. 10.1007/​BF02084158.

[67] Edwin T. Jaynes. Information theory and statistical mechanics. Physical review, 106 (4): 620, 1957. 10.1103/​PhysRev.106.620.

[68] Renato Renner. Symmetry of large physical systems implies independence of subsystems. Nature Physics, 3 (9): 645–649, 2007. 10.1038/​nphys684.

[69] N H Y Ng, L Mančinska, C Cirstoiu, J Eisert, and S Wehner. Limits to catalysis in quantum thermodynamics. New Journal of Physics, 17 (8): 085004, 2015. 10.1088/​1367-2630/​17/​8/​085004.

[70] Wim van Dam and Patrick Hayden. Universal entanglement transformations without communication. Physical Review A, 67 (6): 060302, 2003. 10.1103/​PhysRevA.67.060302.

[71] Karen V. Hovhannisyan, Martí Perarnau-Llobet, Marcus Huber, and Antonio Acín. Entanglement Generation is Not Necessary for Optimal Work Extraction. Physical Review Letters, 111 (24): 240401, 2013. 10.1103/​PhysRevLett.111.240401.

[72] Marcus Huber, Martí Perarnau-Llobet, Karen V. Hovhannisyan, Paul Skrzypczyk, Claude Klöckl, Nicolas Brunner, and Antonio Acín. Thermodynamic cost of creating correlations. New Journal of Physics, 17 (6): 065008, 2015. 10.1088/​1367-2630/​17/​6/​065008.

[73] H. Wilming, R. Gallego, and J. Eisert. Second law of thermodynamics under control restrictions. Physical Review E, 93 (4): 042126, 2016. 10.1103/​PhysRevE.93.042126.

[74] Cédric Bény, Christopher T. Chubb, Terry Farrelly, and Tobias J. Osborne. Energy cost of entanglement extraction in complex quantum systems. Nature Communications, 9 (1): 3792, 2018. 10.1038/​s41467-018-06153-w.

[75] J. Lekscha, H. Wilming, J. Eisert, and R. Gallego. Quantum thermodynamics with local control. Physical Review E, 97 (2): 022142, 2018. 10.1103/​PhysRevE.97.022142.

[76] Vlatko Vedral, Martin B. Plenio, M. A. Rippin, and Peter L. Knight. Quantifying Entanglement. Physical Review Letters, 78 (12): 2275–2279, 1997. 10.1103/​PhysRevLett.78.2275.

[77] Koenraad Audenaert, Jens Eisert, E. Jané, Martin B. Plenio, Shashank Virmani, and Bart De Moor. Asymptotic Relative Entropy of Entanglement. Physical Review Letters, 87 (21): 217902, 2001. 10.1103/​PhysRevLett.87.217902.

[78] Adam Miranowicz and Satoshi Ishizaka. Closed formula for the relative entropy of entanglement. Physical Review A, 78 (3): 032310, 2008. 10.1103/​PhysRevA.78.032310.

[79] Alvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry. Entanglement fluctuation theorems. Physical Review A, 100: 012317, 2019. 10.1103/​PhysRevA.100.012317.

[80] Alvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry. Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality. Physical Review X, 6 (4): 041017, 2016. 10.1103/​PhysRevX.6.041017.

[81] Joseph M. Renes. Relative submajorization and its use in quantum resource theories. Journal of Mathematical Physics, 57 (12): 122202, 2016. 10.1063/​1.4972295.

[82] Benjamin Morris and Gerardo Adesso. Quantum coherence fluctuation relations. Journal of Physics A: Mathematical and Theoretical, 51 (41): 414007, 2018. 10.1088/​1751-8121/​aac115.

[83] Aram W. Harrow. Entanglement spread and clean resource inequalities. In XVIth International Congress on Mathematical Physics, pages 536–540. World Scientific, 2010. 10.1142/​9789814304634_0046.

[84] Robert Alicki and Mark Fannes. Entanglement boost for extractable work from ensembles of quantum batteries. Physical Review E, 87 (4), 2013. 10.1103/​PhysRevE.87.042123.

[85] Johan Åberg. Catalytic Coherence. Physical Review Letters, 113 (15): 150402, 2014. 10.1103/​PhysRevLett.113.150402.

[86] Matteo Lostaglio, David Jennings, and Terry Rudolph. Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nature Communications, 6: 6383, 2015. 10.1038/​ncomms7383.

[87] Hyukjoon Kwon, Hyunseok Jeong, David Jennings, Benjamin Yadin, and M. S. Kim. Clock–Work Trade-Off Relation for Coherence in Quantum Thermodynamics. Physical Review Letters, 120 (15): 150602, 2018. 10.1103/​PhysRevLett.120.150602.

[88] Matthew J. Donald, Michał Horodecki, and Oliver Rudolph. The uniqueness theorem for entanglement measures. Journal of Mathematical Physics, 43 (9): 4252–4272, 2002. 10.1063/​1.1495917.

[89] Matthias Christandl. The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography. arXiv:quant-ph/​0604183, 2006. URL https:/​/​​abs/​quant-ph/​0604183.

[90] Benjamin Schumacher. Quantum coding. Physical Review A, 51 (4): 2738–2747, 1995. 10.1103/​PhysRevA.51.2738.

Cited by

[1] Sandu Popescu, Ana Belén Sainz, Anthony J. Short, and Andreas Winter, "Reference Frames Which Separately Store Noncommuting Conserved Quantities", Physical Review Letters 125 9, 090601 (2020).

[2] Andrés F. Ducuara, Patryk Lipka-Bartosik, and Paul Skrzypczyk, "Multiobject operational tasks for convex quantum resource theories of state-measurement pairs", Physical Review Research 2 3, 033374 (2020).

[3] Gilad Gour and Carlo Maria Scandolo, "Dynamical Entanglement", Physical Review Letters 125 18, 180505 (2020).

[4] Eric Chitambar and Gilad Gour, "Quantum resource theories", Reviews of Modern Physics 91 2, 025001 (2019).

[5] Zi-Wen Liu and Andreas Winter, "Resource theories of quantum channels and the universal role of resource erasure", arXiv:1904.04201.

[6] Zi-Wen Liu, Kaifeng Bu, and Ryuji Takagi, "One-Shot Operational Quantum Resource Theory", Physical Review Letters 123 2, 020401 (2019).

[7] Carlo Maria Scandolo, "Information-theoretic foundations of thermodynamics in general probabilistic theories", arXiv:1901.08054.

[8] Graeme D. Berk, Andrew J. P. Garner, Benjamin Yadin, Kavan Modi, and Felix A. Pollock, "Resource theories of multi-time processes: A window into quantum non-Markovianity", arXiv:1907.07003.

[9] Tomáš Gonda and Robert W. Spekkens, "Monotones in General Resource Theories", arXiv:1912.07085.

[10] Karol Horodecki and Maciej Stankiewicz, "Semi-Device Independent Quantum Money", arXiv:1811.10552, New Journal of Physics 22 2, 023007 (2018).

[11] Nicole Yunger Halpern, Michael E. Beverland, and Amir Kalev, "Noncommuting conserved charges in quantum many-body thermalization", Physical Review E 101 4, 042117 (2020).

[12] Mischa P. Woods and Michał Horodecki, "The Resource Theoretic Paradigm of Quantum Thermodynamics with Control", arXiv:1912.05562.

[13] John H. Selby and Ciarán M. Lee, "Compositional resource theories of coherence", arXiv:1911.04513.

[14] F. H. Kamin, F. T. Tabesh, S. Salimi, and F. Kheirandish, "The resource theory of coherence for quantum channels", Quantum Information Processing 19 7, 210 (2020).

[15] Andrés F. Ducuara, Patryk Lipka-Bartosik, and Paul Skrzypczyk, "Multi-object operational tasks for convex quantum resource theories", arXiv:2004.12898.

The above citations are from Crossref's cited-by service (last updated successfully 2021-01-18 22:02:48) and SAO/NASA ADS (last updated successfully 2021-01-18 22:02:49). The list may be incomplete as not all publishers provide suitable and complete citation data.