# Heralded generation of maximal entanglement in any dimension via incoherent coupling to thermal baths

Armin Tavakoli1, Géraldine Haack1, Marcus Huber2, Nicolas Brunner1, and Jonatan Bohr Brask1

1Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland
2Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria

### Abstract

We present a scheme for dissipatively generating maximal entanglement in a heralded manner. Our setup requires incoherent interactions with two thermal baths at different temperatures, but no source of work or control. A pair of $(d+1)$-dimensional quantum systems is first driven to an entangled steady state by the temperature gradient, and maximal entanglement in dimension $d$ can then be heralded via local filters. We discuss experimental prospects considering an implementation in superconducting systems.

In this work, we show how a thermal machine can be used to create quantum entanglement by coupling to baths at different temperatures and exploiting the heat flows between them.

In the classical, everyday world, thermal machines do a lot of useful tasks for us. Power plants turn heat into electricity. Refrigerators keep our beers cold. Steam locomotives pull trains (OK, at least they used to). In general, they are machines which move heat around, or transform it. Often by connecting different points of the machine to different temperatures and exploiting the resulting heat flows.

Classical thermal machines are big, and generally one does not need to worry about quantum physics to understand what is going on. But what happens if make such a machine smaller and smaller, to the point where quantum effects become important? Say we take a locomotive and scale it down, down, down, until the boiler and gears and so on consist of just a few atoms? Of course, it won't really be a locomotive any more - but maybe we can learn something interesting?

Indeed, we can. And the machine can still be useful.

In recent years, physicists have learned a lot about thermodynamics on the quantum scale by studying such tiny thermal machines. Looking at how fundamental concepts from classical thermodynamics, such as the 2nd law or Carnot efficiency, behave in the quantum regime, we gain new insights into the differences between the classical and quantum worlds.

We can also think about whether there are new tasks that such quantum thermal machines could do.

Entanglement is an essential quantum phenomena. Objects which are entangled behave as if they are a single entity even when separated and manipulated independently. This enables new, powerful applications such as quantum computing and quantum metrology, and is at the heart of the foundations of quantum physics. So creating and studying entanglement is very interesting from both fundamental and applied points of view. Might a thermal machine be used to generate entanglement then?

Entanglement is generally very fragile, and thermal noise tends to wash it out quickly. In fact, a lot of effort in quantum physics experiments goes into keeping the systems cold and isolated, so as to be able to observe the genuinely quantum effects. So it is not at all obvious that using a thermal machine for entanglement generation would work. However, it turns out that connecting with noisy environments can indeed help to create and keep entanglement stable, in certain systems.

This has been studied for a variety of systems and settings. In this paper, we focus on very simple machines which require no external control or driving and use just a temperature difference between hot and cold baths. It was found previously that entanglement can indeed be generated in this setting. However, the amount of entanglement was rather limited. Here, we present a new quantum thermal machine which generates maximal entanglement. It can do this for the simplest possible case of just two quantum bits, but also for two quantum trits, and in fact for two quantum systems of any dimension. The new machine again uses just two different temperatures and two quantum systems. One system is connected to a cold bath, one to a hot bath, and they interact with each other. When heat flows from hot to cold through the two systems, they become entangled. Local filters allow the extraction of maximal entanglement.

Our work opens a path towards thermal generation of larger entangled states useful for building quantum computers or sensitive quantum sensors.

### ► References

[1] M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, Cavity-loss-induced generation of entangled atoms,'' Phys. Rev. A 59, 2468–2475 (1999).
https:/​/​doi.org/​10.1103/​PhysRevA.59.2468

[2] M. S. Kim, Jinhyoung Lee, D. Ahn, and P. L. Knight, Entanglement induced by a single-mode heat environment,'' Phys. Rev. A 65, 040101 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.65.040101

[3] L. Jakóbczyk, Entangling two qubits by dissipation,'' J. Phys. A: Math. Gen. , 6383 (2002).
https:/​/​doi.org/​10.1088/​0305-4470/​35/​30/​313

[4] D. Braun, Creation of entanglement by interaction with a common heat bath,'' Phys. Rev. Lett. 89, 277901 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.277901

[5] F. Benatti, R. Floreanini, and M. Piani, Environment induced entanglement in markovian dissipative dynamics,'' Phys. Rev. Lett. 91, 070402 (2003).
https:/​/​doi.org/​10.1103/​PhysRevLett.91.070402

[6] D. Burgarth and V. Giovannetti, Mediated homogenization,'' Phys. Rev. A 76, 062307 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.76.062307

[7] B. Bellomo, R. Lo Franco, S. Maniscalco, and G. Compagno, Entanglement trapping in structured environments,'' Phys. Rev. A 78, 060302 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.060302

[8] D. Manzano, M. Tiersch, A. Asadian, and H. J. Briegel, Quantum transport efficiency and fourier's law,'' Phys. Rev. E 86, 061118 (2012).
https:/​/​doi.org/​10.1103/​PhysRevE.86.061118

[9] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Buchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms,'' Nat Phys 4, 878–883 (2008).
https:/​/​doi.org/​10.1038/​nphys1073

[10] F. Verstraete, M. M. Wolf, and I. J. Cirac, Quantum computation and quantum-state engineering driven by dissipation,'' Nat Phys 5, 633–636 (2009).
https:/​/​doi.org/​10.1038/​nphys1342

[11] B. Kraus, H. P. Büchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller, Preparation of entangled states by quantum markov processes,'' Phys. Rev. A 78, 042307 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.042307

[12] F. Ticozzi and L. Viola, Steady-state entanglement by engineered quasi-local markovian dissipation,'' Quant. Inf. and Comp. 14, 0265 (2014).
http:/​/​www.rintonpress.com/​xxqic14/​qic-14-34/​0265-0294.pdf

[13] F. Tacchino, A. Auffèves, M. F. Santos, and D. Gerace, Steady state entanglement beyond thermal limits,'' Phys. Rev. Lett. 120, 063604 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.063604

[14] S. Schneider and G. J. Milburn, Entanglement in the steady state of a collective-angular-momentum (dicke) model,'' Phys. Rev. A 65, 042107 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.65.042107

[15] M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Dissipative preparation of entanglement in optical cavities,'' Phys. Rev. Lett. 106, 090502 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.090502

[16] X. Wang and S. G. Schirmer, Generating maximal entanglement between non-interacting atoms by collective decay and symmetry breaking,'' arXiv e-print , 1005.2114 (2010).
arXiv:1005.2114

[17] F. Reiter, L. Tornberg, G. Johansson, and A. S. Sørensen, Steady-state entanglement of two superconducting qubits engineered by dissipation,'' Phys. Rev. A 88, 032317 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.032317

[18] M. J. A. Schuetz, E. M. Kessler, L. M. K. Vandersypen, J. I. Cirac, and G. Giedke, Steady-state entanglement in the nuclear spin dynamics of a double quantum dot,'' Phys. Rev. Lett. 111, 246802 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.246802

[19] J. Cai, S. Popescu, and H. J. Briegel, Dynamic entanglement in oscillating molecules and potential biological implications,'' Phys. Rev. E 82, 021921 (2010).
https:/​/​doi.org/​10.1103/​PhysRevE.82.021921

[20] S. Walter, J. C. Budich, J. Eisert, and B. Trauzettel, Entanglement of nanoelectromechanical oscillators by cooper-pair tunneling,'' Phys. Rev. B 88, 035441 (2013).
https:/​/​doi.org/​10.1103/​PhysRevB.88.035441

[21] H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,'' Phys. Rev. Lett. 107, 080503 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.080503

[22] J. T. Barreiro, M. Muller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, An open-system quantum simulator with trapped ions,'' Nature 470, 486–491 (2011).
https:/​/​doi.org/​10.1038/​nature09801

[23] Y. Lin, J. P. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. S. Sorensen, D. Leibfried, and D. J. Wineland, Dissipative production of a maximally entangled steady state of two quantum bits,'' Nature 504, 415–418 (2013).
https:/​/​doi.org/​10.1038/​nature12801

[24] S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Autonomously stabilized entanglement between two superconducting quantum bits,'' Nature 504, 419–422 (2013).
https:/​/​doi.org/​10.1038/​nature12802

[25] G. Vacanti and A. Beige, Cooling atoms into entangled states,'' New Journal of Physics 11, 083008 (2009).
https:/​/​doi.org/​10.1088/​1367-2630/​11/​8/​083008

[26] F. Reiter, M. J. Kastoryano, and A. S. Sørensen, Driving two atoms in an optical cavity into an entangled steady state using engineered decay,'' New Journal of Physics 14, 053022 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​5/​053022

[27] C. Aron, M. Kulkarni, and H. E. Türeci, Steady-state entanglement of spatially separated qubits via quantum bath engineering,'' Phys. Rev. A 90, 062305 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.062305

[28] M. B. Plenio and S. F. Huelga, Entangled light from white noise,'' Phys. Rev. Lett. 88, 197901 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.197901

[29] L. Hartmann, W. Dür, and H.-J. Briegel, Steady-state entanglement in open and noisy quantum systems,'' Phys. Rev. A 74, 052304 (2006).
https:/​/​doi.org/​10.1103/​PhysRevA.74.052304

[30] L. Hartmann, W. Dür, and H. J. Briegel, Entanglement and its dynamics in open, dissipative systems,'' New Journal of Physics 9, 230 (2007).
https:/​/​doi.org/​10.1088/​1367-2630/​9/​7/​230

[31] L. Quiroga, F. J. Rodríguez, M. E. Ramírez, and R. París, Nonequilibrium thermal entanglement,'' Phys. Rev. A 75, 032308 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.032308

[32] M. Žnidarič, Entanglement in stationary nonequilibrium states at high energies,'' Phys. Rev. A 85, 012324 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.85.012324

[33] B. Bellomo and M. Antezza, Steady entanglement out of thermal equilibrium,'' EPL (Europhysics Letters) 104, 10006 (2013a).
https:/​/​doi.org/​10.1209/​0295-5075/​104/​10006

[34] B. Bellomo and M. Antezza, Creation and protection of entanglement in systems out of thermal equilibrium,'' New Journal of Physics 15, 113052 (2013b).
https:/​/​doi.org/​10.1088/​1367-2630/​15/​11/​113052

[35] D. Boyanovsky and D. Jasnow, Coherence of mechanical oscillators mediated by coupling to different baths,'' Phys. Rev. A 96, 012103 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.012103

[36] J. B. Brask, G. Haack, N. Brunner, and M. Huber, Autonomous quantum thermal machine for generating steady-state entanglement,'' New Journal of Physics 17, 113029 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​11/​113029

[37] N. Linden, S. Popescu, and P. Skrzypczyk, How small can thermal machines be? the smallest possible refrigerator,'' Phys. Rev. Lett. 105, 130401 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.105.130401

[38] P. P. Hofer, M. Perarnau-Llobet, L. D. M. Miranda, G. Haack, R.Silva, J. B. Brask, and N. Brunner, Markovian master equations for quantum thermal machines: local versus global approach,'' New Journal of Physics 19, 123037 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa964f

[39] J. O. González, L. A. Correa, G. Nocerino, J. P. Palao, D. Alonso, and G. Adesso, Testing the Validity of the Local' and Global' GKLS Master Equations on an Exactly Solvable Model,'' Open Systems & Information Dynamics 24, 1740010 (2017).
https:/​/​doi.org/​10.1142/​S1230161217400108

[40] G. Vidal and R. F. Werner, Computable measure of entanglement,'' Phys. Rev. A 65, 032314 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.65.032314

[41] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories,'' Phys. Rev. Lett. 23, 880–884 (1969).
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[42] F. Verstraete, K. Audenaert, J. Dehaene, and B. De Moor, A comparison of the entanglement measures negativity and concurrence,'' J. Phys. A: Math. Gen. 34, 10327 (2001).
https:/​/​doi.org/​10.1088/​0305-4470/​34/​47/​329

[43] M. A. Nielsen, Conditions for a class of entanglement transformations,'' Phys. Rev. Lett. 83, 436–439 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.436

[44] Y.-X. Chen and S.-W. Li, Quantum refrigerator driven by cur-rent noise,'' Europhys. Lett. 97, 40003 (2012).
https:/​/​doi.org/​10.1209/​0295-5075/​97/​40003

[45] P. P. Hofer, J.-R. Souquet, and A. A. Clerk, Quantum heat engine based on photon-assisted cooper pair tunneling,'' Phys. Rev. B 93, 041418 (2016a).
https:/​/​doi.org/​10.1103/​PhysRevB.93.041418

[46] P. P. Hofer, M. Perarnau-Llobet, J. B. Brask, R. Silva, M. Huber, and N. Brunner, Autonomous quantum refrigerator in a circuit qed architecture based on a josephson junction,'' Phys. Rev. B 94, 235420 (2016b).
https:/​/​doi.org/​10.1103/​PhysRevB.94.235420

[47] X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, Microwave photonics with superconducting quantum circuits,'' Physics Reports 718-719, 1 – 102 (2017).
https:/​/​doi.org/​10.1016/​j.physrep.2017.10.002

[48] V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, Fluxonium: Single cooper-pair circuit free of charge offsets,'' Science 326, 113–116 (2009).
https:/​/​doi.org/​10.1126/​science.1175552

[49] G. Zhu, D. G. Ferguson, V. E. Manucharyan, and J. Koch, Circuit qed with fluxonium qubits: Theory of the dispersive regime,'' Phys. Rev. B 87, 024510 (2013).
https:/​/​doi.org/​10.1103/​PhysRevB.87.024510

[50] V.E. Manucharyan, Superinductance, Ph.D. thesis (2012).

[51] I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles,'' Nature 508, 369 (2014).
https:/​/​doi.org/​10.1038/​nature13017

[52] Y.-H. Lin, L. B. Nguyen, N. Grabon, J. San Miguel, N. Pankratova, and V. E. Manucharyan, Demonstration of protection of a superconducting qubit from energy decay,'' Phys. Rev. Lett. 120, 150503 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.150503

[53] J. Majer, J. M. Chow, J. M. Gambetta, Jens Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Coupling superconducting qubits via a cavity bus,'' Nature 449, 443 EP – (2007).
https:/​/​doi.org/​10.1038/​nature06184

[54] Mika A. Sillanpää, Jae I. Park, and Raymond W. Simmonds, Coherent quantum state storage and transfer between two phase qubits via a resonant cavity,'' Nature 449, 438 EP – (2007).
https:/​/​doi.org/​10.1038/​nature06124

[55] L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Demonstration of two-qubit algorithms with a superconducting quantum processor,'' Nature 460, 240 EP – (2009).
https:/​/​doi.org/​10.1038/​nature08121

[56] N. Cottet, Private communication,''.

[57] Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O'Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, Qubit architecture with high coherence and fast tunable coupling,'' Phys. Rev. Lett. 113, 220502 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.220502

[58] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Resolving photon number states in a superconducting circuit,'' Nature 445, 515 EP – (2007).
https:/​/​doi.org/​10.1038/​nature05461

[59] N. Cottet, S. Jezouin, L. Bretheau, P. Campagne-Ibarcq, Q. Ficheux, J. Anders, A. Auffèves, R. Azouit, P. Rouchon, and B. Huard, Observing a quantum maxwell demon at work,'' Proc. Natl. Acad. Sci. U.S.A. 114, 7561–7564 (2017).
https:/​/​doi.org/​10.1073/​pnas.1704827114

[60] M. Jerger, P. Macha, A. R. Hamann, Y. Reshitnyk, K. Juliusson, and A. Fedorov, Realization of a binary-outcome projection measurement of a three-level superconducting quantum system,'' Phys. Rev. Applied 6, 014014 (2016).
https:/​/​doi.org/​10.1103/​PhysRevApplied.6.014014

[61] E. Jeffrey, D. Sank, J. Y. Mutus, T. C. White, J. Kelly, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Megrant, P. J. J. O'Malley, C. Neill, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, Fast accurate state measurement with superconducting qubits,'' Phys. Rev. Lett. 112, 190504 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.190504

[62] N. T. Bronn, Y. Liu, J. B. Hertzberg, A. D. Córcoles, A. A. Houck, J. M. Gambetta, and J. M. Chow, Broadband filters for abatement of spontaneous emission in circuit quantum electrodynamics,'' Applied Physics Letters 107, 172601 (2015).
https:/​/​doi.org/​10.1063/​1.4934867

[63] A. Kou, W. C. Smith, U. Vool, I. M. Pop, K. M. Sliwa, M. H. Hatridge, L. Frunzio, and M. H. Devoret, Simultaneous monitoring of fluxonium qubits in a waveguide,'' arXiv e-print , 1705.05712 (2017).
arXiv:1705.05712

[64] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of nâlevel systems,'' J. Math. Phys. 17, 821–825 (1976).
https:/​/​doi.org/​10.1063/​1.522979

[65] G. Lindblad, On the generators of quantum dynamical semigroups,'' Commun. Math. Phys. 48, 119–130 (1976).
https:/​/​doi.org/​10.1007/​BF01608499

[66] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).

[67] C. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag Berlin Heidelberg, 2004).

[68] G. Schaller, Non-Equilibrium Master Equations (Technische Universität Berlin, 2015).

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[2] Niels Lörch, Christoph Bruder, Nicolas Brunner, and Patrick P Hofer, "Optimal work extraction from quantum states by photo-assisted Cooper pair tunneling", Quantum Science and Technology 3 3, 035014 (2018).

[3] Kun Zhang, Wei Wu, and Jin Wang, "Influence of equilibrium and nonequilibrium environments on macroscopic realism through the Leggett-Garg inequalities", Physical Review A 101 5, 052334 (2020).

[4] Shishir Khandelwal, Nicolas Palazzo, Nicolas Brunner, and Géraldine Haack, "Critical heat current for operating an entanglement engine", New Journal of Physics 22 7, 073039 (2020).

[5] Karen V Hovhannisyan and Alberto Imparato, "Quantum current in dissipative systems", New Journal of Physics 21 5, 052001 (2019).

[6] Mark T. Mitchison, "Quantum thermal absorption machines: refrigerators, engines and clocks", Contemporary Physics 60 2, 164 (2019).

[7] Armin Tavakoli, Géraldine Haack, Nicolas Brunner, and Jonatan Bohr Brask, "Autonomous multipartite entanglement engines", Physical Review A 101 1, 012315 (2020).

[8] 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 (2019).

[9] Zhihai Wang, Wei Wu, and Jin Wang, "Steady-state entanglement and coherence of two coupled qubits in equilibrium and nonequilibrium environments", Physical Review A 99 4, 042320 (2019).

[10] 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 (2017).

[11] Mark T. Mitchison and Martin B. Plenio, "Non-additive dissipation in open quantum networks out of equilibrium", New Journal of Physics 20 3, 033005 (2018).

[12] F. Tacchino, A. Auffèves, M. F. Santos, and D. Gerace, "Steady State Entanglement beyond Thermal Limits", Physical Review Letters 120 6, 063604 (2018).

The above citations are from Crossref's cited-by service (last updated successfully 2020-09-22 19:25:32) and SAO/NASA ADS (last updated successfully 2020-09-22 19:25:33). The list may be incomplete as not all publishers provide suitable and complete citation data.