Operational nonclassicality in minimal autonomous thermal machines

Jonatan Bohr Brask1, Fabien Clivaz2,3, Géraldine Haack4, and Armin Tavakoli3,5

1Department of Physics, Technical University of Denmark, 2800 Kongens Lyngby, Denmark
2Institut für Theoretische Physik und IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
3Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
4Département de Physique Appliqée, Université de Genève, 1211 Geneva, Switzerland
5Atominstitut, Technische Universität Wien, Stadionallee 2, 1020 Vienna, Austria

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

Abstract

Thermal machines exploit interactions with multiple heat baths to perform useful tasks, such as work production and refrigeration. In the quantum regime, tasks with no classical counterpart become possible. Here, we consider the minimal setting for quantum thermal machines, namely two-qubit autonomous thermal machines that use only incoherent interactions with their environment, and investigate the fundamental resources needed to generate entanglement. Our investigation is systematic, covering different types of interactions, bosonic and fermionic environments, and different resources that can be supplied to the machine. We adopt an operational perspective in which we assess the nonclassicality of the generated entanglement through its ability to perform useful tasks such as Einstein-Podolsky-Rosen steering, quantum teleportation and Bell nonlocality. We provide both constructive examples of nonclassical effects and general no-go results that demarcate the fundamental limits in autonomous entanglement generation. Our results open up a path toward understanding nonclassical phenomena in thermal processes.

► BibTeX data

► References

[1] J. S. Bell. On the einstein podolsky rosen paradox. Physics Physique Fizika, 1: 195–200, Nov 1964. 10.1103/​PhysicsPhysiqueFizika.1.195. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysicsPhysiqueFizika.1.195.
https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195

[2] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Phys. Rev. Lett., 70: 1895–1899, Mar 1993. 10.1103/​PhysRevLett.70.1895. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.70.1895.
https:/​/​doi.org/​10.1103/​PhysRevLett.70.1895

[3] Sourav Bhattacharjee and Amit Dutta. Quantum thermal machines and batteries, 2020. https:/​/​doi.org/​10.1140/​epjb/​s10051-021-00235-3.
https:/​/​doi.org/​10.1140/​epjb/​s10051-021-00235-3

[4] Felix Binder, Luis A. Correa, Christian Gogolin, Janet Anders, and Gerardo Adesso, editors. Thermodynamics in the Quantum Regime. Springer, Cham, 2018. https:/​/​doi.org/​10.1007/​978-3-319-99046-0.
https:/​/​doi.org/​10.1007/​978-3-319-99046-0

[5] Jonatan Bohr Brask, Géraldine Haack, Nicolas Brunner, and Marcus Huber. Autonomous quantum thermal machine for generating steady-state entanglement. New Journal of Physics, 17 (11): 113029, nov 2015. 10.1088/​1367-2630/​17/​11/​113029. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​11/​113029

[6] Heinz-Peter Breuer and Francesco Petruccione, editors. The theory of open quantum systems. Oxford University Press, 2010. https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001

[7] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419–478, Apr 2014a. 10.1103/​RevModPhys.86.419. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.86.419.
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[8] Nicolas Brunner, Marcus Huber, Noah Linden, Sandu Popescu, Ralph Silva, and Paul Skrzypczyk. Entanglement enhances cooling in microscopic quantum refrigerators. Phys. Rev. E, 89: 032115, Mar 2014b. 10.1103/​PhysRevE.89.032115. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevE.89.032115.
https:/​/​doi.org/​10.1103/​PhysRevE.89.032115

[9] Marco Cattaneo, Gian Luca Giorgi, Sabrina Maniscalco, and Roberta Zambrini. Local versus global master equation with common and separate baths: superiority of the global approach in partial secular approximation. New Journal of Physics, 21 (11): 113045, nov 2019. 10.1088/​1367-2630/​ab54ac. URL https:/​/​doi.org/​10.1088/​1367-2630/​ab54ac.
https:/​/​doi.org/​10.1088/​1367-2630/​ab54ac

[10] D Cavalcanti and P Skrzypczyk. Quantum steering: a review with focus on semidefinite programming. Reports on Progress in Physics, 80 (2): 024001, dec 2016. 10.1088/​1361-6633/​80/​2/​024001. URL https:/​/​doi.org/​10.1088/​1361-6633/​80/​2/​024001.
https:/​/​doi.org/​10.1088/​1361-6633/​80/​2/​024001

[11] Gabriele De Chiara, Gabriel Landi, Adam Hewgill, Brendan Reid, Alessandro Ferraro, Augusto J Roncaglia, and Mauro Antezza. Reconciliation of quantum local master equations with thermodynamics. New Journal of Physics, 20 (11): 113024, nov 2018. 10.1088/​1367-2630/​aaecee. URL https:/​/​doi.org/​10.1088/​1367-2630/​aaecee.
https:/​/​doi.org/​10.1088/​1367-2630/​aaecee

[12] B. S. Cirel'son. Quantum generalizations of Bell's inequality. Lett. Math. Phys., 4 (2): 93–100, Mar 1980. ISSN 1573-0530. 10.1007/​BF00417500. URL https:/​/​doi.org/​10.1007/​BF00417500.
https:/​/​doi.org/​10.1007/​BF00417500

[13] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23: 880–884, Oct 1969. 10.1103/​PhysRevLett.23.880. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.23.880.
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[14] Fabien Clivaz, Ralph Silva, Géraldine Haack, Jonatan Bohr Brask, Nicolas Brunner, and Marcus Huber. Unifying paradigms of quantum refrigeration: A universal and attainable bound on cooling. Phys. Rev. Lett., 123: 170605, Oct 2019. 10.1103/​PhysRevLett.123.170605. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.123.170605.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.170605

[15] 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? Phys. Rev. X, 7: 031022, Aug 2017. 10.1103/​PhysRevX.7.031022. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.7.031022.
https:/​/​doi.org/​10.1103/​PhysRevX.7.031022

[16] D. Gelbwaser-Klimovsky and G. Kurizki. Heat-machine control by quantum-state preparation: From quantum engines to refrigerators. Phys. Rev. E, 90: 022102, Aug 2014. 10.1103/​PhysRevE.90.022102. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevE.90.022102.
https:/​/​doi.org/​10.1103/​PhysRevE.90.022102

[17] David Gelbwaser-Klimovsky, Alexei Bylinskii, Dorian Gangloff, Rajibul Islam, Alán Aspuru-Guzik, and Vladan Vuletic. Single-atom heat machines enabled by energy quantization. Phys. Rev. Lett., 120: 170601, Apr 2018. 10.1103/​PhysRevLett.120.170601. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.120.170601.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.170601

[18] N. Gisin. Bell's inequality holds for all non-product states. Physics Letters A, 154 (5): 201–202, 1991. ISSN 0375-9601. https:/​/​doi.org/​10.1016/​0375-9601(91)90805-I. URL https:/​/​www.sciencedirect.com/​science/​article/​pii/​037596019190805I.
https:/​/​doi.org/​10.1016/​0375-9601(91)90805-I
https:/​/​www.sciencedirect.com/​science/​article/​pii/​037596019190805I

[19] J. Onam González, Luis A. Correa, Giorgio Nocerino, José P. Palao, Daniel Alonso, and Gerardo Adesso. Testing the validity of the ‘local’ and ‘global’ gkls master equations on an exactly solvable model. Open Systems & Information Dynamics, 24 (04): 1740010, 2017. 10.1142/​S1230161217400108. URL https:/​/​doi.org/​10.1142/​S1230161217400108.
https:/​/​doi.org/​10.1142/​S1230161217400108

[20] Scott Hill and William K. Wootters. Entanglement of a pair of quantum bits. Phys. Rev. Lett., 78: 5022–5025, Jun 1997. 10.1103/​PhysRevLett.78.5022. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.78.5022.
https:/​/​doi.org/​10.1103/​PhysRevLett.78.5022

[21] Patrick P Hofer, Martí Perarnau-Llobet, L David M Miranda, Géraldine Haack, Ralph Silva, Jonatan Bohr Brask, and Nicolas Brunner. Markovian master equations for quantum thermal machines: local versus global approach. New Journal of Physics, 19 (12): 123037, dec 2017. 10.1088/​1367-2630/​aa964f. URL https:/​/​doi.org/​10.1088/​1367-2630/​aa964f.
https:/​/​doi.org/​10.1088/​1367-2630/​aa964f

[22] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. General teleportation channel, singlet fraction, and quasidistillation. Phys. Rev. A, 60: 1888–1898, Sep 1999. 10.1103/​PhysRevA.60.1888. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.60.1888.
https:/​/​doi.org/​10.1103/​PhysRevA.60.1888

[23] R. Horodecki, P. Horodecki, and M. Horodecki. Violating bell inequality by mixed spin-12 states: necessary and sufficient condition. Physics Letters A, 200 (5): 340–344, 1995. ISSN 0375-9601. https:/​/​doi.org/​10.1016/​0375-9601(95)00214-N. URL https:/​/​www.sciencedirect.com/​science/​article/​pii/​037596019500214N.
https:/​/​doi.org/​10.1016/​0375-9601(95)00214-N
https:/​/​www.sciencedirect.com/​science/​article/​pii/​037596019500214N

[24] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81: 865–942, Jun 2009. 10.1103/​RevModPhys.81.865. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.81.865.
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[25] A. Iorio, E. Strambini, G. Haack, M. Campisi, and F. Giazotto. Photonic heat rectification in a system of coupled qubits. Phys. Rev. Applied, 15: 054050, May 2021. 10.1103/​PhysRevApplied.15.054050. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevApplied.15.054050.
https:/​/​doi.org/​10.1103/​PhysRevApplied.15.054050

[26] Martin Josefsson, Artis Svilans, Adam M. Burke, Eric A. Hoffmann, Sofia Fahlvik, Claes Thelander, Martin Leijnse, and Heiner Linke. A quantum-dot heat engine operating close to the thermodynamic efficiency limits. Nature Nanotechnology, 13 (10): 920–924, Oct 2018. ISSN 1748-3395. 10.1038/​s41565-018-0200-5. URL https:/​/​doi.org/​10.1038/​s41565-018-0200-5.
https:/​/​doi.org/​10.1038/​s41565-018-0200-5

[27] Shishir Khandelwal, Nicolas Palazzo, Nicolas Brunner, and Géraldine Haack. Critical heat current for operating an entanglement engine. New Journal of Physics, 22 (7): 073039, jul 2020. 10.1088/​1367-2630/​ab9983. URL https:/​/​doi.org/​10.1088/​1367-2630/​ab9983.
https:/​/​doi.org/​10.1088/​1367-2630/​ab9983

[28] Amikam Levy and David Gelbwaser-Klimovsky. Quantum Features and Signatures of Quantum Thermal Machines, pages 87–126. Springer International Publishing, Cham, 2018. ISBN 978-3-319-99046-0. 10.1007/​978-3-319-99046-0_4. URL https:/​/​doi.org/​10.1007/​978-3-319-99046-0_4.
https:/​/​doi.org/​10.1007/​978-3-319-99046-0_4

[29] Amikam Levy and Ronnie Kosloff. Quantum absorption refrigerator. Phys. Rev. Lett., 108: 070604, Feb 2012. 10.1103/​PhysRevLett.108.070604. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.108.070604.
https:/​/​doi.org/​10.1103/​PhysRevLett.108.070604

[30] Noah Linden, Sandu Popescu, and Paul Skrzypczyk. How small can thermal machines be? the smallest possible refrigerator. Phys. Rev. Lett., 105: 130401, Sep 2010. 10.1103/​PhysRevLett.105.130401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.105.130401.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.130401

[31] Zhong-Xiao Man, Armin Tavakoli, Jonatan Bohr Brask, and Yun-Jie Xia. Improving autonomous thermal entanglement generation using a common reservoir. Physica Scripta, 94 (7): 075101, apr 2019. 10.1088/​1402-4896/​ab0c51. URL https:/​/​doi.org/​10.1088/​1402-4896/​ab0c51.
https:/​/​doi.org/​10.1088/​1402-4896/​ab0c51

[32] Gleb Maslennikov, Shiqian Ding, Roland Hablützel, Jaren Gan, Alexandre Roulet, Stefan Nimmrichter, Jibo Dai, Valerio Scarani, and Dzmitry Matsukevich. Quantum absorption refrigerator with trapped ions. Nature Communications, 10 (1): 202, Jan 2019. ISSN 2041-1723. 10.1038/​s41467-018-08090-0. URL https:/​/​doi.org/​10.1038/​s41467-018-08090-0.
https:/​/​doi.org/​10.1038/​s41467-018-08090-0

[33] Mark T. Mitchison. Quantum thermal absorption machines: refrigerators, engines and clocks. Contemporary Physics, 60 (2): 164–187, 2019. 10.1080/​00107514.2019.1631555. URL https:/​/​doi.org/​10.1080/​00107514.2019.1631555.
https:/​/​doi.org/​10.1080/​00107514.2019.1631555

[34] Mark T Mitchison and Martin B Plenio. Non-additive dissipation in open quantum networks out of equilibrium. New Journal of Physics, 20 (3): 033005, mar 2018. 10.1088/​1367-2630/​aa9f70. URL https:/​/​doi.org/​10.1088/​1367-2630/​aa9f70.
https:/​/​doi.org/​10.1088/​1367-2630/​aa9f70

[35] H. Chau Nguyen, Huy-Viet Nguyen, and Otfried Gühne. Geometry of einstein-podolsky-rosen correlations. Phys. Rev. Lett., 122: 240401, Jun 2019. 10.1103/​PhysRevLett.122.240401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.122.240401.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.240401

[36] S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein. Advances in quantum teleportation. Nature Photonics, 9 (10): 641–652, Oct 2015. ISSN 1749-4893. 10.1038/​nphoton.2015.154. URL https:/​/​doi.org/​10.1038/​nphoton.2015.154.
https:/​/​doi.org/​10.1038/​nphoton.2015.154

[37] Archak Purkayastha, Abhishek Dhar, and Manas Kulkarni. Out-of-equilibrium open quantum systems: A comparison of approximate quantum master equation approaches with exact results. Phys. Rev. A, 93: 062114, Jun 2016. 10.1103/​PhysRevA.93.062114. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.93.062114.
https:/​/​doi.org/​10.1103/​PhysRevA.93.062114

[38] Johannes Roßnagel, Samuel T. Dawkins, Karl N. Tolazzi, Obinna Abah, Eric Lutz, Ferdinand Schmidt-Kaler, and Kilian Singer. A single-atom heat engine. Science, 352 (6283): 325–329, 2016. ISSN 0036-8075. 10.1126/​science.aad6320. URL https:/​/​science.sciencemag.org/​content/​352/​6283/​325.
https:/​/​doi.org/​10.1126/​science.aad6320
https:/​/​science.sciencemag.org/​content/​352/​6283/​325

[39] Alexandre Roulet, Stefan Nimmrichter, Juan Miguel Arrazola, Stella Seah, and Valerio Scarani. Autonomous rotor heat engine. Phys. Rev. E, 95: 062131, Jun 2017. 10.1103/​PhysRevE.95.062131. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevE.95.062131.
https:/​/​doi.org/​10.1103/​PhysRevE.95.062131

[40] D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde. Experimental epr-steering using bell-local states. Nature Physics, 6 (11): 845–849, Nov 2010. ISSN 1745-2481. 10.1038/​nphys1766. URL https:/​/​doi.org/​10.1038/​nphys1766.
https:/​/​doi.org/​10.1038/​nphys1766

[41] Jorden Senior, Azat Gubaydullin, Bayan Karimi, Joonas T. Peltonen, Joachim Ankerhold, and Jukka P. Pekola. Heat rectification via a superconducting artificial atom. Communications Physics, 3 (1): 40, Feb 2020. ISSN 2399-3650. 10.1038/​s42005-020-0307-5. URL https:/​/​doi.org/​10.1038/​s42005-020-0307-5.
https:/​/​doi.org/​10.1038/​s42005-020-0307-5

[42] Ivan Šupić and Joseph Bowles. Self-testing of quantum systems: a review. Quantum, 4: 337, September 2020. ISSN 2521-327X. 10.22331/​q-2020-09-30-337. URL https:/​/​doi.org/​10.22331/​q-2020-09-30-337.
https:/​/​doi.org/​10.22331/​q-2020-09-30-337

[43] F. Tacchino, A. Auffèves, M. F. Santos, and D. Gerace. Steady state entanglement beyond thermal limits. Phys. Rev. Lett., 120: 063604, Feb 2018. 10.1103/​PhysRevLett.120.063604. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.120.063604.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.063604

[44] Armin Tavakoli and Nicolas Gisin. The Platonic solids and fundamental tests of quantum mechanics. Quantum, 4: 293, July 2020. ISSN 2521-327X. 10.22331/​q-2020-07-09-293. URL https:/​/​doi.org/​10.22331/​q-2020-07-09-293.
https:/​/​doi.org/​10.22331/​q-2020-07-09-293

[45] Armin Tavakoli and Roope Uola. Measurement incompatibility and steering are necessary and sufficient for operational contextuality. Phys. Rev. Research, 2: 013011, Jan 2020. 10.1103/​PhysRevResearch.2.013011. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.2.013011.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.013011

[46] Armin Tavakoli, Alastair A. Abbott, Marc-Olivier Renou, Nicolas Gisin, and Nicolas Brunner. Semi-device-independent characterization of multipartite entanglement of states and measurements. Phys. Rev. A, 98: 052333, Nov 2018a. 10.1103/​PhysRevA.98.052333. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.98.052333.
https:/​/​doi.org/​10.1103/​PhysRevA.98.052333

[47] Armin Tavakoli, Géraldine Haack, Marcus Huber, Nicolas Brunner, and Jonatan Bohr Brask. Heralded generation of maximal entanglement in any dimension via incoherent coupling to thermal baths. Quantum, 2: 73, June 2018b. ISSN 2521-327X. 10.22331/​q-2018-06-13-73. URL https:/​/​doi.org/​10.22331/​q-2018-06-13-73.
https:/​/​doi.org/​10.22331/​q-2018-06-13-73

[48] Armin Tavakoli, Géraldine Haack, Nicolas Brunner, and Jonatan Bohr Brask. Autonomous multipartite entanglement engines. Phys. Rev. A, 101: 012315, Jan 2020. 10.1103/​PhysRevA.101.012315. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.101.012315.
https:/​/​doi.org/​10.1103/​PhysRevA.101.012315

[49] Armin Tavakoli, Alejandro Pozas-Kerstjens, mingxing luo, and Marc-Olivier Renou. Bell nonlocality in networks. Reports on Progress in Physics, 2021. URL http:/​/​iopscience.iop.org/​article/​10.1088/​1361-6633/​ac41bb. 10.1088/​1361-6633/​ac41bb.
https:/​/​doi.org/​10.1088/​1361-6633/​ac41bb

[50] Friedemann Tonner and Günter Mahler. Autonomous quantum thermodynamic machines. Phys. Rev. E, 72: 066118, Dec 2005. 10.1103/​PhysRevE.72.066118. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevE.72.066118.
https:/​/​doi.org/​10.1103/​PhysRevE.72.066118

[51] Roope Uola, Ana C. S. Costa, H. Chau Nguyen, and Otfried Gühne. Quantum steering. Rev. Mod. Phys., 92: 015001, Mar 2020. 10.1103/​RevModPhys.92.015001. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.92.015001.
https:/​/​doi.org/​10.1103/​RevModPhys.92.015001

[52] Frank Verstraete, Koenraad Audenaert, Jeroen Dehaene, and Bart De Moor. A comparison of the entanglement measures negativity and concurrence. Journal of Physics A: Mathematical and General, 34 (47): 10327–10332, 2001. 10.1088/​0305-4470/​34/​47/​329.
https:/​/​doi.org/​10.1088/​0305-4470/​34/​47/​329

[53] G. Vidal and R. F. Werner. Computable measure of entanglement. Phys. Rev. A, 65: 032314, Feb 2002. 10.1103/​PhysRevA.65.032314. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.65.032314.
https:/​/​doi.org/​10.1103/​PhysRevA.65.032314

[54] S. Virmani and M.B. Plenio. Ordering states with entanglement measures. Physics Letters A, 268 (1): 31–34, 2000. ISSN 0375-9601. https:/​/​doi.org/​10.1016/​S0375-9601(00)00157-2. URL https:/​/​www.sciencedirect.com/​science/​article/​pii/​S0375960100001572.
https:/​/​doi.org/​10.1016/​S0375-9601(00)00157-2
https:/​/​www.sciencedirect.com/​science/​article/​pii/​S0375960100001572

[55] Zhihai Wang, Wei Wu, and Jin Wang. Steady-state entanglement and coherence of two coupled qubits in equilibrium and nonequilibrium environments. Phys. Rev. A, 99: 042320, Apr 2019. 10.1103/​PhysRevA.99.042320. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.99.042320.
https:/​/​doi.org/​10.1103/​PhysRevA.99.042320

[56] Reinhard F. Werner. Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable model. Phys. Rev. A, 40: 4277–4281, Oct 1989. 10.1103/​PhysRevA.40.4277. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.40.4277.
https:/​/​doi.org/​10.1103/​PhysRevA.40.4277

[57] H. M. Wiseman, S. J. Jones, and A. C. Doherty. Steering, entanglement, nonlocality, and the einstein-podolsky-rosen paradox. Phys. Rev. Lett., 98: 140402, Apr 2007. 10.1103/​PhysRevLett.98.140402. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.98.140402.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.140402

[58] Mischa P. Woods, Ralph Silva, and Jonathan Oppenheim. Autonomous quantum machines and finite-sized clocks. Annales Henri Poincaré, 20 (1): 125–218, Jan 2019. ISSN 1424-0661. 10.1007/​s00023-018-0736-9. URL https:/​/​doi.org/​10.1007/​s00023-018-0736-9.
https:/​/​doi.org/​10.1007/​s00023-018-0736-9

[59] William K. Wootters. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett., 80: 2245–2248, Mar 1998. 10.1103/​PhysRevLett.80.2245. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.80.2245.
https:/​/​doi.org/​10.1103/​PhysRevLett.80.2245

Cited by

[1] Bradley Longstaff and Jonatan Bohr Brask, "Persistent nonlocality in an ultracold-atom environment", arXiv:2203.04727.

The above citations are from SAO/NASA ADS (last updated successfully 2022-05-28 19:12:09). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2022-05-28 19:12:07).