Minimal energy cost of entanglement extraction

Lucas Hackl1,2 and Robert H. Jonsson3

1Max Planck Institute of Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
2Munich Center for Quantum Science and Technology, Schellingstraße 4, D-80799 München, Germany
3QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

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We compute the minimal energy cost for extracting entanglement from the ground state of a bosonic or fermionic quadratic system. Specifically, we find the minimal energy increase in the system resulting from replacing an entangled pair of modes, sharing entanglement entropy $\Delta S$, by a product state, and we show how to construct modes achieving this minimal energy cost. Thus, we obtain a protocol independent lower bound on the extraction of pure state entanglement from quadratic systems. Due to their generality, our results apply to a large range of physical systems, as we discuss with examples.

Entanglement is one of the most exciting phenomena of quantum physics. If different parts of a system are entangled, they are intimately (cor)-related with each other. Even when they are far separated from each other, measurements on one part of the system can reveal the state of the other part. Over the last decades physicists came up with intriguing applications of entanglement, such as quantum teleportation: Here the full state of, e.g., a qubit gets transferred from place to another while only a single classical bit (0 or 1) of information is sent from A to B.
For such applications, the two parties need entangled quantum systems ready in their labs. So how can such entangled resources be obtained or created? One solution could be to extract the entanglement from other systems. Excitingly, the ground states of many physical systems (such as ultra-cold atoms or quantum fields) are highly entangled. However, extracting this entanglement to our laboratory is not for free but incurs an energy cost. This is because in the process we inevitably change the state of the system and raise its energy.
What our research addresses is how to find the cheapest way possible to extract a given amount of entanglement from such systems. It is like searching for the cheapest gas station in the city, just that we now buy entanglement and pay with energy. What we find is that extracting larger amounts of entanglement becomes increasingly expensive--as if the first gallon of gas only cost $1, but the second gallon already cost 2 and third even 4. This means if you only need to extract a little bit of entanglement, you can land a bargain, but if you need a lot it will be very expensive.
What makes our results appealing is that they are very general, applying to a wide range of different systems including cold atoms and quantum fields. Moreover, our findings are quantitative and concrete, which means we can tell you the exact cost of entanglement and even how you need to couple your lab system to the entangled system to pay this low price. And even if an experimenter cannot exactly follow our recommended procedure, due to experimental limitations, she can at least estimate how far away she is from the optimal price. This is useful in the same way as it is good to know the cheapest gas price in the neighborhood to judge if the gas station across the street offers a good, decent or bad deal.

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[1] A. Einstein, B. Podolsky, and N. Rosen. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev., 47 (10): 777–780, May 1935. 10.1103/​PhysRev.47.777. URL https:/​/​​doi/​10.1103/​PhysRev.47.777.

[2] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1 (3): 195–200, November 1964. 10.1103/​PhysicsPhysiqueFizika.1.195. URL https:/​/​​doi/​10.1103/​PhysicsPhysiqueFizika.1.195.

[3] J. S. Bell. On the einstein podolsky rosen paradox. In John S Bell on the Foundations of Quantum Mechanics, pages 7–12. WORLD SCIENTIFIC, August 2001. ISBN 978-981-02-4687-7. 10.1142/​9789812386540_0002. URL https:/​/​​doi/​abs/​10.1142/​9789812386540_0002.

[4] B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526 (7575): 682–686, October 2015. ISSN 1476-4687. 10.1038/​nature15759. URL https:/​/​​articles/​nature15759.

[5] Marissa Giustina, Marijn A. M. Versteegh, Sören Wengerowsky, Johannes Handsteiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler, Jan-Åke Larsson, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Morgan W. Mitchell, Jörn Beyer, Thomas Gerrits, Adriana E. Lita, Lynden K. Shalm, Sae Woo Nam, Thomas Scheidl, Rupert Ursin, Bernhard Wittmann, and Anton Zeilinger. Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons. Phys. Rev. Lett., 115 (25): 250401, December 2015. 10.1103/​PhysRevLett.115.250401. URL https:/​/​​doi/​10.1103/​PhysRevLett.115.250401.

[6] Lynden K. Shalm, Evan Meyer-Scott, Bradley G. Christensen, Peter Bierhorst, Michael A. Wayne, Martin J. Stevens, Thomas Gerrits, Scott Glancy, Deny R. Hamel, Michael S. Allman, Kevin J. Coakley, Shellee D. Dyer, Carson Hodge, Adriana E. Lita, Varun B. Verma, Camilla Lambrocco, Edward Tortorici, Alan L. Migdall, Yanbao Zhang, Daniel R. Kumor, William H. Farr, Francesco Marsili, Matthew D. Shaw, Jeffrey A. Stern, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Thomas Jennewein, Morgan W. Mitchell, Paul G. Kwiat, Joshua C. Bienfang, Richard P. Mirin, Emanuel Knill, and Sae Woo Nam. Strong Loophole-Free Test of Local Realism. Phys. Rev. Lett., 115 (25): 250402, December 2015. 10.1103/​PhysRevLett.115.250402. URL https:/​/​​doi/​10.1103/​PhysRevLett.115.250402.

[7] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81 (2): 865–942, June 2009. 10.1103/​RevModPhys.81.865. URL https:/​/​​doi/​10.1103/​RevModPhys.81.865.

[8] Charles H. Bennett and Stephen J. Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., 69 (20): 2881–2884, November 1992. 10.1103/​PhysRevLett.69.2881. URL https:/​/​​doi/​10.1103/​PhysRevLett.69.2881.

[9] 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 (13): 1895–1899, March 1993. 10.1103/​PhysRevLett.70.1895. URL https:/​/​​doi/​10.1103/​PhysRevLett.70.1895.

[10] Charles H. Bennett and Gilles Brassard. Quantum cryptography: Public key distribution and coin tossing. Theoretical Computer Science, 560: 7–11, December 2014. ISSN 0304-3975. 10.1016/​j.tcs.2014.05.025. URL http:/​/​​science/​article/​pii/​S0304397514004241.

[11] Mark Srednicki. Entropy and area. Phys. Rev. Lett., 71 (5): 666–669, August 1993. 10.1103/​PhysRevLett.71.666. URL https:/​/​​doi/​10.1103/​PhysRevLett.71.666.

[12] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev. Entanglement in Quantum Critical Phenomena. Phys. Rev. Lett., 90 (22): 227902, June 2003. 10.1103/​PhysRevLett.90.227902. URL https:/​/​​doi/​10.1103/​PhysRevLett.90.227902.

[13] Tobias J. Osborne and Michael A. Nielsen. Entanglement in a simple quantum phase transition. Phys. Rev. A, 66 (3): 032110, September 2002. 10.1103/​PhysRevA.66.032110. URL https:/​/​​doi/​10.1103/​PhysRevA.66.032110.

[14] J. Eisert, M. Cramer, and M. B. Plenio. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys., 82 (1): 277–306, February 2010. 10.1103/​RevModPhys.82.277. URL https:/​/​​doi/​10.1103/​RevModPhys.82.277.

[15] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys., 80 (2): 517–576, May 2008. 10.1103/​RevModPhys.80.517. URL https:/​/​​doi/​10.1103/​RevModPhys.80.517.

[16] Rajibul Islam, Ruichao Ma, Philipp M. Preiss, M. Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. Measuring entanglement entropy in a quantum many-body system. Nature, 528 (7580): 77–83, December 2015. ISSN 1476-4687. 10.1038/​nature15750. URL https:/​/​​articles/​nature15750.

[17] Shinsei Ryu and Tadashi Takayanagi. Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/​Conformal Field Theory Correspondence. Phys. Rev. Lett., 96 (18): 181602, May 2006. 10.1103/​PhysRevLett.96.181602. URL https:/​/​​doi/​10.1103/​PhysRevLett.96.181602.

[18] Stephen J. Summers and Reinhard Werner. The vacuum violates Bell's inequalities. Physics Letters A, 110 (5): 257–259, July 1985. ISSN 0375-9601. 10.1016/​0375-9601(85)90093-3. URL http:/​/​​science/​article/​pii/​0375960185900933.

[19] Stephen J. Summers and Reinhard Werner. Bell's inequalities and quantum field theory. II. Bell's inequalities are maximally violated in the vacuum. J. Math. Phys., 28 (10): 2448–2456, October 1987a. ISSN 0022-2488, 1089-7658. 10.1063/​1.527734. URL http:/​/​​content/​aip/​journal/​jmp/​28/​10/​10.1063/​1.527734.

[20] Stephen J. Summers and Reinhard Werner. Bell's inequalities and quantum field theory. I. General setting. Journal of Mathematical Physics, 28 (10): 2440–2447, October 1987b. ISSN 0022-2488. 10.1063/​1.527733. URL http:/​/​​doi/​abs/​10.1063/​1.527733.

[21] Antony Valentini. Non-local correlations in quantum electrodynamics. Physics Letters A, 153 (6): 321–325, March 1991. ISSN 0375-9601. 10.1016/​0375-9601(91)90952-5. URL http:/​/​​science/​article/​pii/​0375960191909525.

[22] Benni Reznik, Alex Retzker, and Jonathan Silman. Violating Bell's inequalities in vacuum. Phys. Rev. A, 71 (4): 042104, April 2005. 10.1103/​PhysRevA.71.042104. URL http:/​/​​doi/​10.1103/​PhysRevA.71.042104.

[23] Grant Salton, Robert B. Mann, and Nicolas C. Menicucci. Acceleration-assisted entanglement harvesting and rangefinding. New J. Phys., 17 (3): 035001, 2015. ISSN 1367-2630. 10.1088/​1367-2630/​17/​3/​035001. URL http:/​/​​1367-2630/​17/​i=3/​a=035001.

[24] Greg Ver Steeg and Nicolas C. Menicucci. Entangling power of an expanding universe. Phys. Rev. D, 79 (4): 044027, February 2009. 10.1103/​PhysRevD.79.044027. URL http:/​/​​doi/​10.1103/​PhysRevD.79.044027.

[25] M. Cliche and A. Kempf. Vacuum entanglement enhancement by a weak gravitational field. Phys. Rev. D, 83 (4): 045019, February 2011. 10.1103/​PhysRevD.83.045019. URL http:/​/​​doi/​10.1103/​PhysRevD.83.045019.

[26] Eduardo Martín-Martínez, Alexander R. H. Smith, and Daniel R. Terno. Spacetime structure and vacuum entanglement. Phys. Rev. D, 93 (4): 044001, February 2016. 10.1103/​PhysRevD.93.044001. URL https:/​/​​doi/​10.1103/​PhysRevD.93.044001.

[27] Daniel Braun. Creation of Entanglement by Interaction with a Common Heat Bath. Phys. Rev. Lett., 89 (27): 277901, December 2002. 10.1103/​PhysRevLett.89.277901. URL https:/​/​​doi/​10.1103/​PhysRevLett.89.277901.

[28] Daniel Braun. Entanglement from thermal blackbody radiation. Phys. Rev. A, 72 (6): 062324, December 2005. 10.1103/​PhysRevA.72.062324. URL https:/​/​​doi/​10.1103/​PhysRevA.72.062324.

[29] Eric G. Brown. Thermal amplification of field-correlation harvesting. Phys. Rev. A, 88 (6): 062336, December 2013. 10.1103/​PhysRevA.88.062336. URL http:/​/​​doi/​10.1103/​PhysRevA.88.062336.

[30] Allison Sachs, Robert B. Mann, and Eduardo Martín-Martínez. Entanglement harvesting and divergences in quadratic Unruh-DeWitt detector pairs. Phys. Rev. D, 96 (8): 085012, October 2017. 10.1103/​PhysRevD.96.085012. URL https:/​/​​doi/​10.1103/​PhysRevD.96.085012.

[31] Petar Simidzija, Robert H. Jonsson, and Eduardo Martín-Martínez. General no-go theorem for entanglement extraction. Phys. Rev. D, 97 (12): 125002, June 2018. 10.1103/​PhysRevD.97.125002. URL https:/​/​​doi/​10.1103/​PhysRevD.97.125002.

[32] Petar Simidzija and Eduardo Martin-Martinez. Harvesting correlations from thermal and squeezed coherent states. Phys. Rev. D, 98 (8): 085007, October 2018. ISSN 2470-0010, 2470-0029. 10.1103/​PhysRevD.98.085007. URL http:/​/​​abs/​1809.05547.

[33] S. Jay Olson and Timothy C. Ralph. Entanglement between the Future and the Past in the Quantum Vacuum. Phys. Rev. Lett., 106 (11): 110404, March 2011. 10.1103/​PhysRevLett.106.110404. URL https:/​/​​doi/​10.1103/​PhysRevLett.106.110404.

[34] S. Jay Olson and Timothy C. Ralph. Extraction of timelike entanglement from the quantum vacuum. Phys. Rev. A, 85 (1): 012306, January 2012. 10.1103/​PhysRevA.85.012306. URL https:/​/​​doi/​10.1103/​PhysRevA.85.012306.

[35] Carlos Sabín, Borja Peropadre, Marco del Rey, and Eduardo Martín-Martínez. Extracting Past-Future Vacuum Correlations Using Circuit QED. Phys. Rev. Lett., 109 (3): 033602, July 2012. 10.1103/​PhysRevLett.109.033602. URL https:/​/​​doi/​10.1103/​PhysRevLett.109.033602.

[36] Alejandro Pozas-Kerstjens and Eduardo Martín-Martínez. Entanglement harvesting from the electromagnetic vacuum with hydrogenlike atoms. Phys. Rev. D, 94 (6): 064074, September 2016. 10.1103/​PhysRevD.94.064074. URL https:/​/​​doi/​10.1103/​PhysRevD.94.064074.

[37] Fernando Galve and Eric Lutz. Energy cost and optimal entanglement production in harmonic chains. Phys. Rev. A, 79 (3): 032327, March 2009. 10.1103/​PhysRevA.79.032327. URL https:/​/​​doi/​10.1103/​PhysRevA.79.032327.

[38] Eduardo Martín-Martínez, Eric G. Brown, William Donnelly, and Achim Kempf. Sustainable entanglement production from a quantum field. Phys. Rev. A, 88 (5): 052310, November 2013. 10.1103/​PhysRevA.88.052310. URL http:/​/​​doi/​10.1103/​PhysRevA.88.052310.

[39] Marcus Huber, Martí Perarnau-Llobet, Karen V. Hovhannisyan, Paul Skrzypczyk, Claude Klöckl, Nicolas Brunner, and Antonio Acín. Thermodynamic cost of creating correlations. New J. Phys., 17 (6): 065008, June 2015. ISSN 1367-2630. 10.1088/​1367-2630/​17/​6/​065008. URL https:/​/​​10.1088%2F1367-2630%2F17%2F6%2F065008.

[40] Nicolai Friis, Marcus Huber, and Martí Perarnau-Llobet. Energetics of correlations in interacting systems. Phys. Rev. E, 93 (4): 042135, April 2016. 10.1103/​PhysRevE.93.042135. URL https:/​/​​doi/​10.1103/​PhysRevE.93.042135.

[41] Tamoghna Das, Asutosh Kumar, Amit Kumar Pal, Namrata Shukla, Aditi Sen(De), and Ujjwal Sen. Canonical distillation of entanglement. Physics Letters A, 381 (41): 3529–3535, November 2017. ISSN 0375-9601. 10.1016/​j.physleta.2017.08.065. URL http:/​/​​science/​article/​pii/​S0375960117308393.

[42] Giulio Chiribella and Yuxiang Yang. Optimal quantum operations at zero energy cost. Phys. Rev. A, 96 (2): 022327, August 2017. 10.1103/​PhysRevA.96.022327. URL https:/​/​​doi/​10.1103/​PhysRevA.96.022327.

[43] Giuseppe Vitagliano, Claude Klöckl, Marcus Huber, and Nicolai Friis. Trade-Off Between Work and Correlations in Quantum Thermodynamics. In Felix Binder, Luis A. Correa, Christian Gogolin, Janet Anders, and Gerardo Adesso, editors, Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, Fundamental Theories of Physics, pages 731–750. Springer International Publishing, Cham, 2018. ISBN 978-3-319-99046-0. 10.1007/​978-3-319-99046-0_30. URL https:/​/​​10.1007/​978-3-319-99046-0_30.

[44] Masahiro Hotta. Quantum measurement information as a key to energy extraction from local vacuums. Phys. Rev. D, 78 (4): 045006, August 2008. 10.1103/​PhysRevD.78.045006. URL http:/​/​​doi/​10.1103/​PhysRevD.78.045006.

[45] Masahiro Hotta. Controlled Hawking process by quantum energy teleportation. Phys. Rev. D, 81 (4): 044025, February 2010a. 10.1103/​PhysRevD.81.044025. URL http:/​/​​doi/​10.1103/​PhysRevD.81.044025.

[46] Masahiro Hotta. Energy entanglement relation for quantum energy teleportation. Physics Letters A, 374 (34): 3416–3421, July 2010b. ISSN 0375-9601. 10.1016/​j.physleta.2010.06.058. URL http:/​/​​science/​article/​pii/​S0375960110007723.

[47] Masahiro Hotta, Jiro Matsumoto, and Go Yusa. Quantum energy teleportation without a limit of distance. Phys. Rev. A, 89 (1): 012311, January 2014. 10.1103/​PhysRevA.89.012311. URL http:/​/​​doi/​10.1103/​PhysRevA.89.012311.

[48] Cédric Bény, Christopher T. Chubb, Terry Farrelly, and Tobias J. Osborne. Energy cost of entanglement extraction in complex quantum systems. Nat. Commun., 9 (1): 3792, September 2018. ISSN 2041-1723. 10.1038/​s41467-018-06153-w. URL https:/​/​​articles/​s41467-018-06153-w.

[49] Ted Jacobson. Entanglement Equilibrium and the Einstein Equation. Phys. Rev. Lett., 116 (20), May 2016. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.116.201101. URL http:/​/​​abs/​1505.04753.

[50] M. Hotta, R. Schützhold, and W. G. Unruh. On the partner particles for moving mirror radiation and black hole evaporation. ArXiv150306109 Gr-Qc Physicsquant-Ph, March 2015. URL http:/​/​​abs/​1503.06109.

[51] Jose Trevison, Koji Yamaguchi, and Masahiro Hotta. Spatially overlapped partners in quantum field theory. J. Phys. A: Math. Theor., 52 (12): 125402, February 2019. ISSN 1751-8121. 10.1088/​1751-8121/​ab065b. URL https:/​/​​10.1088%2F1751-8121%2Fab065b.

[52] Ashtekar A., Magnon Anne, and Penrose Roger. Quantum fields in curved space-times. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 346 (1646): 375–394, November 1975. 10.1098/​rspa.1975.0181. URL https:/​/​​doi/​abs/​10.1098/​rspa.1975.0181.

[53] Robert M. Wald. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics. University of Chicago Press, Chicago, 1994. ISBN 0-226-87025-1.

[54] Alonso Botero and Benni Reznik. Mode-Wise Entanglement of Gaussian States. Phys. Rev. A, 67 (5), May 2003. ISSN 1050-2947, 1094-1622. 10.1103/​PhysRevA.67.052311. URL http:/​/​​abs/​quant-ph/​0209026.

[55] Michael M. Wolf. Not-So-Normal Mode Decomposition. Phys. Rev. Lett., 100 (7): 070505, February 2008. 10.1103/​PhysRevLett.100.070505. URL https:/​/​​doi/​10.1103/​PhysRevLett.100.070505.

[56] Eugenio Bianchi, Jonathan Guglielmon, Lucas Hackl, and Nelson Yokomizo. Squeezed vacua in loop quantum gravity. ArXiv160505356 Gr-Qc Physicshep-Th, May 2016. URL http:/​/​​abs/​1605.05356.

[57] Lucas Hackl, Eugenio Bianchi, Ranjan Modak, and Marcos Rigol. Entanglement production in bosonic systems: Linear and logarithmic growth. Phys. Rev. A, 97 (3): 032321, March 2018. 10.1103/​PhysRevA.97.032321. URL https:/​/​​doi/​10.1103/​PhysRevA.97.032321.

[58] Lucas Fabian Hackl. Aspects of Gaussian States: Entanglement, Squeezing and Complexity. PhD thesis, Pennsylvania State University, July 2018. URL https:/​/​​catalog/​15815lfh109.

[59] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. Gaussian quantum information. Rev. Mod. Phys., 84 (2): 621–669, May 2012. 10.1103/​RevModPhys.84.621. URL http:/​/​​doi/​10.1103/​RevModPhys.84.621.

[60] M. B. Plenio, J. Eisert, J. Dreißig, and M. Cramer. Entropy, Entanglement, and Area: Analytical Results for Harmonic Lattice Systems. Phys. Rev. Lett., 94 (6): 060503, February 2005. 10.1103/​PhysRevLett.94.060503. URL https:/​/​​doi/​10.1103/​PhysRevLett.94.060503.

[61] H. Casini and M. Huerta. Entanglement entropy in free quantum field theory. J. Phys. A: Math. Theor., 42 (50): 504007, 2009. ISSN 1751-8121. 10.1088/​1751-8113/​42/​50/​504007. URL http:/​/​​1751-8121/​42/​i=50/​a=504007.

[62] Peter Woit. Quantum Theory, Groups and Representations: An Introduction. Springer International Publishing, 2017. ISBN 978-3-319-64610-7. 10.1007/​978-3-319-64612-1. URL https:/​/​​gp/​book/​9783319646107.

[63] R. L. Hudson. When is the wigner quasi-probability density non-negative? Reports on Mathematical Physics, 6 (2): 249–252, October 1974. ISSN 0034-4877. 10.1016/​0034-4877(74)90007-X. URL http:/​/​​science/​article/​pii/​003448777490007X.

[64] Francisco Soto and Pierre Claverie. When is the Wigner function of multidimensional systems nonnegative? Journal of Mathematical Physics, 24 (1): 97–100, January 1983. ISSN 0022-2488. 10.1063/​1.525607. URL https:/​/​​doi/​10.1063/​1.525607.

[65] G. C. Wick. The Evaluation of the Collision Matrix. Phys. Rev., 80 (2): 268–272, October 1950. 10.1103/​PhysRev.80.268. URL https:/​/​​doi/​10.1103/​PhysRev.80.268.

[66] Lev Vidmar, Lucas Hackl, Eugenio Bianchi, and Marcos Rigol. Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians. Phys. Rev. Lett., 119 (2): 020601, July 2017. 10.1103/​PhysRevLett.119.020601. URL https:/​/​​doi/​10.1103/​PhysRevLett.119.020601.

[67] Lev Vidmar, Lucas Hackl, Eugenio Bianchi, and Marcos Rigol. Volume Law and Quantum Criticality in the Entanglement Entropy of Excited Eigenstates of the Quantum Ising Model. Phys. Rev. Lett., 121 (22): 220602, November 2018. 10.1103/​PhysRevLett.121.220602. URL https:/​/​​doi/​10.1103/​PhysRevLett.121.220602.

[68] Lucas Hackl, Lev Vidmar, Marcos Rigol, and Eugenio Bianchi. Average eigenstate entanglement entropy of the XY chain in a transverse field and its universality for translationally invariant quadratic fermionic models. Phys. Rev. B, 99 (7): 075123, February 2019. 10.1103/​PhysRevB.99.075123. URL https:/​/​​doi/​10.1103/​PhysRevB.99.075123.

[69] Shira Chapman, Jens Eisert, Lucas Hackl, Michal Heller, Ro Jefferson, Hugo Marrochio, and Robert Myers. Complexity and entanglement for thermofield double states. SciPost Phys., 6 (3): 034, March 2019. ISSN 2542-4653. 10.21468/​SciPostPhys.6.3.034. URL https:/​/​​10.21468/​SciPostPhys.6.3.034.

[70] Rafael D. Sorkin. On the Entropy of the Vacuum outside a Horizon. ArXiv14023589 Cond-Mat Physicsgr-Qc Physicshep-Th Physicsquant-Ph, February 2014. URL http:/​/​​abs/​1402.3589.

[71] Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D. Sorkin. Quantum source of entropy for black holes. Phys. Rev. D, 34 (2): 373–383, July 1986. 10.1103/​PhysRevD.34.373. URL https:/​/​​doi/​10.1103/​PhysRevD.34.373.

[72] Mari-Carmen Bañuls, J. Ignacio Cirac, and Michael M. Wolf. Entanglement in fermionic systems. Phys. Rev. A, 76 (2), August 2007. ISSN 1050-2947, 1094-1622. 10.1103/​PhysRevA.76.022311. URL http:/​/​​abs/​0705.1103.

[73] Franz Schwabl. Advanced Quantum Mechanics. Springer-Verlag, Berlin Heidelberg, 4 edition, 2008. ISBN 978-3-540-85061-8. 10.1007/​978-3-540-85062-5. URL https:/​/​​gp/​book/​9783540850618.

[74] P. Jordan and E. P. Wigner. Über das Paulische Äquivalenzverbot. In Arthur S. Wightman, editor, The Collected Works of Eugene Paul Wigner: Part A: The Scientific Papers, The Collected Works of Eugene Paul Wigner, pages 109–129. Springer Berlin Heidelberg, Berlin, Heidelberg, 1993. ISBN 978-3-662-02781-3. 10.1007/​978-3-662-02781-3_9. URL https:/​/​​10.1007/​978-3-662-02781-3_9.

[75] Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Annals of Physics, 16 (3): 407–466, December 1961. ISSN 0003-4916. 10.1016/​0003-4916(61)90115-4. URL http:/​/​​science/​article/​pii/​0003491661901154.

[76] T. Holstein and H. Primakoff. Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet. Phys. Rev., 58 (12): 1098–1113, December 1940. 10.1103/​PhysRev.58.1098. URL https:/​/​​doi/​10.1103/​PhysRev.58.1098.

[77] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol. One dimensional bosons: From condensed matter systems to ultracold gases. Rev. Mod. Phys., 83 (4): 1405–1466, December 2011. 10.1103/​RevModPhys.83.1405. URL https:/​/​​doi/​10.1103/​RevModPhys.83.1405.

[78] Pierre Pfeuty. The one-dimensional Ising model with a transverse field. Annals of Physics, 57 (1): 79–90, March 1970. ISSN 0003-4916. 10.1016/​0003-4916(70)90270-8. URL http:/​/​​science/​article/​pii/​0003491670902708.

[79] Oscar C. O. Dahlsten, Cosmo Lupo, Stefano Mancini, and Alessio Serafini. Entanglement typicality. J. Phys. A: Math. Theor., 47 (36): 363001, August 2014. ISSN 1751-8121. 10.1088/​1751-8113/​47/​36/​363001. URL https:/​/​​10.1088%2F1751-8113%2F47%2F36%2F363001.

[80] Lucas Hackl and Robert H. Jonsson. in preparation.

[81] Alex W. Chin, Ángel Rivas, Susana F. Huelga, and Martin B. Plenio. Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials. Journal of Mathematical Physics, 51 (9): 092109, September 2010. ISSN 0022-2488. 10.1063/​1.3490188. URL https:/​/​​doi/​abs/​10.1063/​1.3490188.

[82] Stefano Mancini, Roberto Pierini, and Mark M. Wilde. Preserving information from the beginning to the end of time in a Robertson–Walker spacetime. New J. Phys., 16 (12): 123049, December 2014. ISSN 1367-2630. 10.1088/​1367-2630/​16/​12/​123049. URL https:/​/​​10.1088%2F1367-2630%2F16%2F12%2F123049.

[83] Robert H. Jonsson, Eduardo Martín-Martínez, and Achim Kempf. Information Transmission Without Energy Exchange. Phys. Rev. Lett., 114 (11): 110505, March 2015. 10.1103/​PhysRevLett.114.110505. URL http:/​/​​doi/​10.1103/​PhysRevLett.114.110505.

[84] Robert H. Jonsson. Information travels in massless fields in 1+1 dimensions where energy cannot. J. Phys. A: Math. Theor., 49 (44): 445402, 2016. ISSN 1751-8121. 10.1088/​1751-8113/​49/​44/​445402. URL http:/​/​​1751-8121/​49/​i=44/​a=445402.

[85] Robert H. Jonsson. Quantum signaling in relativistic motion and across acceleration horizons. J. Phys. A: Math. Theor., 50 (35): 355401, 2017. ISSN 1751-8121. 10.1088/​1751-8121/​aa7d3c. URL http:/​/​​1751-8121/​50/​i=35/​a=355401.

[86] Robert H. Jonsson, Eduardo Martín-Martínez, and Achim Kempf. Quantum signaling in cavity QED. Phys. Rev. A, 89 (2): 022330, February 2014. 10.1103/​PhysRevA.89.022330. URL http:/​/​​doi/​10.1103/​PhysRevA.89.022330.

[87] Robert H. Jonsson, Katja Ried, Eduardo Martín-Martínez, and Achim Kempf. Transmitting qubits through relativistic fields. J. Phys. A: Math. Theor., 51 (48): 485301, 2018. ISSN 1751-8121. 10.1088/​1751-8121/​aae78a. URL http:/​/​​1751-8121/​51/​i=48/​a=485301.

[88] Ralph Abraham, Jerrold E Marsden, and Jerrold E Marsden. Foundations of mechanics, volume 36. Benjamin/​Cummings Publishing Company Reading, Massachusetts, 1978.

Cited by

[1] Bennet Windt, Alexander Jahn, Jens Eisert, and Lucas Hackl, "Local optimization on pure Gaussian state manifolds", SciPost Physics 10 3, 066 (2021).

[2] Ivan Romualdo, Lucas Hackl, and Nelson Yokomizo, "Entanglement production in the dynamical Casimir effect at parametric resonance", Physical Review D 100 6, 065022 (2019).

[3] Lucas Hackl and Eugenio Bianchi, "Bosonic and fermionic Gaussian states from Kähler structures", SciPost Physics Core 4 3, 025 (2021).

[4] Lucas Hackl, Tommaso Guaita, Tao Shi, Jutho Haegeman, Eugene Demler, and Ignacio Cirac, "Geometry of variational methods: dynamics of closed quantum systems", SciPost Physics 9 4, 048 (2020).

[5] Hugo A. Camargo, Lucas Hackl, Michal P. Heller, Alexander Jahn, Tadashi Takayanagi, and Bennet Windt, "Entanglement and complexity of purification in ( 1+1 )-dimensional free conformal field theories", Physical Review Research 3 1, 013248 (2021).

[6] Nicolò Piccione, Benedetto Militello, Anna Napoli, and Bruno Bellomo, "Generation of minimum-energy entangled states", Physical Review A 103 6, 062402 (2021).

[7] Koji Yamaguchi, Aida Ahmadzadegan, Petar Simidzija, Achim Kempf, and Eduardo Martín-Martínez, "Superadditivity of channel capacity through quantum fields", Physical Review D 101 10, 105009 (2020).

[8] Robert H. Jonsson, Lucas Hackl, and Krishanu Roychowdhury, "Entanglement dualities in supersymmetry", Physical Review Research 3 2, 023213 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-19 21:29:24). The list may be incomplete as not all publishers provide suitable and complete citation data.

On SAO/NASA ADS no data on citing works was found (last attempt 2021-10-19 21:29:24).