無中文摘要

Two topics on entanglement are concerned in this thesis: One is the decoherence patterns of a topological qubit made of two Majorana zero modes in the generic linear and circular motions and the other is a simplified protocol of quantum energy teleportation (QET) for the holographic conformal field theory in three-dimensional anti-de Sitter space with or without a black hole.
On the first topic, the exact reduced dynamics without Markov approximation is shown. For general time scale, the acceleration causes thermalization as expected by Unruh effect. However, for the short-time scale, the rate
of decoherence is anti-correlated with the acceleration, as a kind of decoherence impedance. This is in fact related to the “anti-Unruh" phenomenon previously found by studying the transition probability of Unruh-DeWitt detector. Besides, the information backflow is observed by some time modulations of coupling constant or acceleration. Moreover, it also shows that some incoherent accelerations of the constituent Majorana zero modes can preserve the coherence instead of thermalizing it.
On the second topic, as a tentative proposal, the standard QET is simplified by replacing Alice’s local measurement with the local projection. At the same time, Bob’s local operation of the usual QET for extracting energy is mimicked by deforming the UV surface with a local bump. Adopting the
surface-state duality, this deformation corresponds to local unitary. In this protocol, the extraction energy is always positive. Moreover, the ratio of extraction energy to the injection one is an universal function of the UV surface deformation profile.

Bibliography
[1] Pei-Hua Liu and Feng-Li Lin, “Decoherence of topological qubit in linear and circular motions: decoherence impedance, anti-Unruh and informa- tion backflow,” JHEP 1607 (2016) 084 [arXiv:1603.05136 [quant-ph]].
[2] Dimitrios Giataganas, Feng-Li Lin, and Pei-Hua Liu , "Towards holo- graphic quantum energy teleportation," Phys. Rev. D 94, 126013 (2016) [arXiv:1608.06523 [hep-th]].
[3] Shih-Hao Ho, Sung-Po Chao ,Chung-Hsien Chou and Feng-Li Lin, “De- coherence patterns of topological qubits from Majorana modes," New J. Phys. 16, no. 11, 113062 (2014) [arXiv:1406.6249 [cond-mat.str-el]].
[4] H. D. Zeh,“On the Interpretation of Measurement in Quantum Theory,” Found. Phys.1: 69 (1970).
[5] W. H. Zurek, Pointer Basis of Quantum Apparatus: Into What Mixture Does the Wave Packet Collapse?, Phys. Rev. D 24, 1516 (1981).
W. H. Zurek, Environment induced superselection rules, Phys. Rev. D 26, 1862 (1982).
[6] W. H. Zurek, “Decoherence and the transition from quantum to classi- cal,” Physics Today 44 (10) 36-44 (1991).
W. H. Zurek,“Decoherence and the transition from quantum to classical
- Revisited,” Los Alamos Science 27, 86-109 (2002).
W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Reviews of Modern Physics, 75, 715-765 (2003).
[7] W. H. Zurek,“Preferred states, predictabilty, classicality, and the envi- ronmentinduced decoherence,” Prog. Theor. Phys. 89, 281312 (1993).
[8] W. H. Zurek, S. Habib, J. P. Paz,“Coherent states via decoherence,” Phys. Rev. Lett. 70, 11871190 (1993).
[9] S. -H. Ho, W. Li, F. -L. Lin and B. Ning,“QuantumDecoherence with Holography," JHEP 1401, 170 (2014). [arXiv:1309.5855 [hep-th]].
[10] R. P. Feynman and F. L. Vernon, Jr., “The theory of a general quantum system interacting with a linear dissipative system, Annals Phys. 24, 118 (1963).
[11] J. S. Schwinger, “Brownian motion of a quantum oscillator,” J. Math.
Phys. 2, 407 (1961).
L. V. Keldysh, “Diagram technique for nonequilibrium processes,” Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)].
[12] Z. -b. Su, L. -y. Chen, X. -t. Yu and K. -c. Chou,“Influence functional and closed-time-path Green’s function,” Phys. Rev. B 37, 9810 (1988).
[13] R. Kubo,“Statistical mechanical theory of irreversible processes. 1. Gen- eral theory and simple applications in magnetic and conduction prob- lems,” J. Phys. Soc. Jap. 12, 570 (1957). P. C. Martin and J. S. Schwinger,“Theory of many particle systems. 1.,” Phys. Rev. 115, 1342 (1959).
[14] A. Einstein,“Uber die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen,” Annalen der Physik 17, 549 (1905).
[15] M. Smoluchowski,“Zur kinetischen Theorie der Brownschen Molekular- bewegung und der Suspensionen,” Annalen der Physik 21, 756 (1906).
[16] P. Langevin,“ Sur la theorie du mouvement brownien,” Comptes rendus de l’Academie des Sciences (Paris), 146, 530 (1908).
[17] A. O. Caldeira and A. J. Leggett,“Influence of dissipation on quantum tunneling in macroscopic systems,” Phys. Rev. Lett. 46, 211(1981).
[18] A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587(1983).
[19] B. L. Hu, J. P. Paz and Y. -h. Zhang, “Quantum Brownian motion in a general environment: 1. Exact master equation with nonlocal dissipa- tion and colored noise,” Phys. Rev. D 45, 2843 (1992).
[20] W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D 14, 870 (1976).
[21] B. S. DeWitt, “Quantum gravity: the new synthesis," in General rela- tivity: An Einstein centenary survey, edited by S. W. Hawking and W. Israel, Cambridge University Press, Cambridge, 680-745 (1979).
[22] E. G. Brown, E. Martin-Martinez, N. C. Menicucci and R. B. Mann, “Detectors for probing relativistic quantum physics beyond perturbation theory,” Phys. Rev. D 87, 084062 (2013) [arXiv:1212.1973 [quant-ph]].
[23] A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires," Phys.
Usp. 44, 131 (2001) [arXiv:cond-mat/0010440].
[24] N. Read and D. Green,“Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect,” Phys. Rev. B 61, 10267 (2000).
[25] G. ’t Hooft, “Dimensional Reduction in Quantum Gravity,” (1993) [arXiv:gr-qc/9310026].
[26] L. Susskind, “The World as a Hologram,” Journal of Mathematical Physics. 36(11): 6377-6396(1995)[arXiv:hep-th/9409089].
[27] J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)][arXiv:hep-th/9711200].
[28] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory cor- relators from non-critical string theory,” Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109].
E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150].
[29] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement en- tropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006) [arXiv:hep- th/0603001].
[30] S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement En- tropy,” JHEP 0608 (2006) 045 [arXiv:hep-th/0605073].
[31] A. Lewkowycz and J. M. Maldacena, “Generalized gravitational en- tropy,”JHEP 1308 (2013) 090 [arXiv:1304.4926 [hep-th]].
[32] A. Kitaev and J. Preskill, “Topological entanglement entropy,” Phys.
Rev. Lett. 96, 110404 (2006).

[33] G. Vidal, “Class of Quantum Many-Body States That Can Be Efficiently Simulated,” Phys. Rev. Lett. 101, 110501 (2008).
[34] G. Vidal, “Entanglement renormalization,” Phys. Rev. Lett. 99, 220405 (2007).
[35] B. Swingle, “Entanglement Renormalization and Holography,” Phys.
Rev. D 86, 065007 (2012) [arXiv:0905.1317 [cond-mat.str-el]].
[36] S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. 69, 2863 (1992).
[37] S. Rommer, S. Ostlund, “A class of ansatz wave functions for 1D spin systems and their relation to DMRG,” Phys. Rev. Lett. 75, 3537 (1995).
[38] F. Verstraete and J.I. Cirac, “ Renormalization Algorithms for Quantum-Many Body Systems in Two and Higher Dimensions,” arXiv:0407066[cond-mat](2004).
[39] G. Evenbly and G. Vidal, “Scaling of entanglement entropy in the (branching) multiscale entanglement renormalization ansatz,” Phys.Rev. B 89 (2014) 235113, [arXiv:1310.8372].
[40] J. Haegeman, T. J. Osborne, H. Verschelde and F. Verstraete, “En- tanglement Renormalization for Quantum Fields in Real Space,” Phys. Rev. Lett. 110, no. 10, 100402 (2013) [arXiv:1102.5524 [hep-th]].
[41] M. Miyaji and T. Takayanagi, “Surface/State Correspondence as a Generalized Holography,” PTEP 2015 (2015) no.7, 073B03 [arXiv:1503.03542 [hep-th]].
[42] M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, “Continuous Multiscale Entanglement Renormalization Ansatz as Holo- graphic Surface-State Correspondence,” Phys. Rev. Lett. 115, no. 17, 171602 (2015) [arXiv:1506.01353 [hep-th]].
[43] M. Nozaki, S. Ryu and T. Takayanagi, “Holographic Geometry of En- tanglement Renormalization in Quantum Field Theories,” JHEP 1210 (2012) 193 [arXiv:1208.3469[hep-th]].
[44] L. Fu and C. L. Kane, “Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator," Phys. Rev. Lett. 100, 096407 (2008).
[45] M. Z. Hasan, C. L. Kane, “Topological Insulators," Rev. Mod. Phys. 82, 3045 (2010).
[46] X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors", Rev. Mod. Phys. 83, 1057 (2011).
[47] M. W.-Y. Tu and W.-M. Zhang, “Non-Markovian decoherence the- ory for a double-dot charge qubit", Phys. Rev. B 78, 235311 (2008) [arXiv:0809.3490 [cond-mat.meshall]].
J. S. Jin, M. W.-Y. Tu, W.-M. Zhang and Y. J. Yan, “Non-equilibrium quantum theory for nanodevices based on the Feynman Vernon influence functional", New J. Phys. 12 083013 (2010) [arXiv:0910.1675 [cond- mat.meshall]].
[48] H.-B. Liu, J.-H. An, C. Chen, Q.-J. Tong, H.-G. Luo and C. H. Oh,
“Anomalous decoherence in a dissipative two-level system", Phys. Rev. A 87, 052139 (2013).
[49] S. T. Wu, “Quenched decoherence in qubit dynamics due to strong amplitude-damping noise”, Phys. Rev. A 89,034301 (2014) [arXiv:1310.6843[quant-ph]].
[50] J. Polchinski, “String theory. Vol. 1: An introduction to the bosonic string,” Cambridge University Press, 2007.
[51] S.-P. Chao, S. A. Silotri, and C.-H. Chung, “Nonequilibrium transport of helical Luttinger liquids through a quantum dot," Phys. Rev. B. 88, 085109 (2013).
[52] S. -H. Ho, W. Li, F. -L. Lin and B. Ning, “Quantum Decoherence with Holography," JHEP 1401, 170 (2014) [arXiv:1309.5855 [hep-th]].
[53] D. T. Son and A. O. Starinets, “Minkowski space correlators in AdS/CFT correspondence: Recipe and applications," JHEP 0209, 042 (2002) [arXiv:hep-th/0205051].
[54] S. Hill, W. K. Wootters, “Entanglement of a Pair of Quantum Bits," Phys. Rev. Lett. 78, 5022 (1997).
W. K. Wootters, “Entanglement of a Pair of Quantum Bits", Phys. Rev. Lett. 80, 2245 (1998).
[55] L. C. B. Crispino, A. Higuchi and G. E. A. Matsas, “The Unruh effect and its applications,” Rev. Mod. Phys. 80, 787 (2008) [arXiv:0710.5373 [gr-qc]].
[56] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math.
Phys. 43, 199 (1975) [Commun. Math. Phys. 46, 206 (1976)].
[57] A. Peres and D. R. Terno, “Quantum information and relativity theory,” Rev. Mod. Phys. 76, 93 (2004) [quant-ph/0212023].
[58] P. M. Alsing and G. J. Milburn, “Teleportation with a Uniformly Ac- celerated Partner," Phys. Rev. Lett. 91, 180404 (2003).
P. M. Alsing, D. McMahon, and G. J. Milburn, “Teleportation in a non- inertial frame," J. Opt. B: Quantum Semiclass. Opt. 6 (2004), S834.
[59] I. Fuentes-Schuller and R. B. Mann, “Alice falls into a black hole: En- tanglement in non-inertial frames,” Phys. Rev. Lett. 95, 120404 (2005) [quant-ph/0410172].
[60] P. M. Alsing, I. Fuentes-Schuller, R. B. Mann and T. E. Tessier, “Entan- glement of Dirac fields in non-inertial frames,” Phys. Rev. A 74, 032326 (2006) [quant-ph/0603269].
[61] S. Y. Lin, C. H. Chou and B. L. Hu, “Disentanglement of two har- monic oscillators in relativistic motion,” Phys. Rev. D 78, 125025 (2008) [arXiv:0803.3995 [gr-qc]].
[62] E. Martin-Martinez, “Relativistic Quantum Information: develop- ments in Quantum Information in general relativistic scenarios,” arXiv:1106.0280 [quant-ph].
[63] D. C. M. Ostapchuk, S. Y. Lin, R. B. Mann and B. L. Hu, “Entanglement Dynamics between Inertial and Non-uniformly Accelerated Detectors,” JHEP 1207, 072 (2012) [arXiv:1108.3377 [gr-qc]].
[64] B. Richter and Y. Omar, “Degradation of entanglement between two accelerated parties: Bell states under the Unruh effect,” Phys. Rev. A 92, 2, 022334 (2015) [arXiv:1503.07526 [quant-ph]].
[65] S. Y. Lin and B. L. Hu, “Backreaction and the Unruh effect: New insights from exact solutions of uniformly accelerated detectors," Phys. Rev. D, 76, 064008 (2007) [arXiv:gr-qc/0611062].
[66] J. S. Bell and J. M. Leinaas, “Electrons As Accelerated Thermometers,” Nucl. Phys. B 212, 131 (1983).
[67] J. S. Bell and J. M. Leinaas, “The Unruh Effect and Quantum Fluctu- ations of Electrons in Storage Rings,” Nucl. Phys. B 284, 488 (1987).
[68] J. R. Letaw and J. D. Pfautsch, “The Quantized Scalar Field in Rotating Coordinates,” Phys. Rev. D 22, 1345 (1980). “The Quantized Scalar Field in the Stationary Coordinate Systems of Flat Space-time,” Phys. Rev. D 24, 1491 (1981).
J. R. Letaw, “Vacuum Excitation of Noninertial Detectors on Stationary World Lines,” Phys. Rev. D 23, 1709 (1981).
P. C. W. Davies, T. Dray and C. A. Manogue, “The Rotating quantum vacuum,” Phys. Rev. D 53, 4382 (1996) [gr-qc/9601034].
O. Levin, Y. Peleg and A. Peres, “Unruh effect for circular motion in a cavity,” J. Phys. A 26, 3001 (1993).
[69] J. Doukas, S. Y. Lin, B. L. Hu and R. B. Mann, “Unruh Effect under Non-equilibrium conditions: Oscillatory motion of an Unruh-DeWitt detector,” JHEP 1311, 119 (2013) [arXiv:1307.4360].
[70] D. Kothawala and T. Padmanabhan, “Response of Unruh-DeWitt de- tector with time-dependent acceleration,” Phys. Lett. B 690, 201 (2010) [arXiv:0911.1017 [gr-qc]].
[71] N. Obadia and M. Milgrom, “On the Unruh effect for general trajecto- ries,” Phys. Rev. D 75, 065006 (2007) [gr-qc/0701130 [GR-QC]].
[72] L. C. Barbado and M. Visser, “Unruh-DeWitt detector event rate for trajectories with time-dependent acceleration,” Phys. Rev. D 86, 084011 (2012) [arXiv:1207.5525 [gr-qc]].
[73] B. F. Svaiter and N. F. Svaiter, “Inertial and noninertial particle detec- tors and vacuum fluctuations,” Phys. Rev. D 46, 5267 (1992).
[74] W. G. Brenna, R. B. Mann and E. Martin-Martinez, “Anti-Unruh Phe- nomena,” Phys. Lett. B 757, 307 (2016) [arXiv:1504.02468 [quant-ph]].
[75] J. Alicea, “Majorana fermions in a tunable semiconductor device," Phys.
Rev. B81, 125318 (2010).
[76] L. Fidkowski, J. Alicea, N. Lindner, R. M. Lutchyn, and M. P.
A. Fisher, “Universal transport signatures of Majorana fermions in superconductor-Luttinger liquid junctions," Phys. Rev. B 85, 245121 (2012) [arXiv:1203.4818 [cond-mat.str-el]].
[77] M. Z. Hasan, C. L. Kane,“Topological Insulators," Rev. Mod. Phys. 82, 3045 (2010).
[78] X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors," Rev. Mod. Phys. 83, 1057 (2011).
[79] Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen, “Symmetry-Protected Topological Orders in Interacting Bosonic Sys- tems," Science 338, 1604 (2012) [arXiv:1301.0861[cond-mat.str-el]].
[80] G. Goldstein, C. Chamon, “Decay rates for topological memories en- coded with Majorana fermions," Phys. Rev. B 85, 205109 (2011) [arXiv:1107.0288 [cond-mat.mes-hall]].
[81] Matisse W. Y. Tu and Wei-Min Zhang , "A non-Markovian decoherence theory for double dot charge qubit," Phys. Rev. B 78, 235311 (2008).
[82] W.-M. Zhang, P.-Y. Lo, H.-N. Xiong, M. W.-Y. Tu, F. Nori, “General non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109, 170402 (2012) [arXiv:1206.4490 [quant-ph]].
[83] D. Hmmer, E. Martin-Martinez and A. Kempf, “Renormalized Unruh- DeWitt Particle Detector Models for Boson and Fermion Fields,” Phys. Rev. D 93, no. 2, 024019 (2016) [arXiv:1506.02046 [quant-ph]].
[84] T.S. Evans, D.A. Steer, "Wick’s theorem at finite temperature," Nucl.
Phys B 474, 481-496 (1996).
[85] H.-P. Breuer, E.-M. Laine, J. Piilo, “Measure for the Degree of Non- Markovian Behavior of Quantum Processes in Open Systems," Phys. Rev. Lett. 103, 210401 (2009).
E.-M. Laine, J. Piilo, H.-P. Breuer, “Measure for the non-Markovianity of quantum processes,"
[86] A. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and Non- Markovianity of Quantum Evolutions," Phys. Rev. Lett. 105, 050403 (2010).
[87] V. Mukhanov and S. Winitzki, “Introduction to quantum effects in grav- ity,” Cambridge University Press (2007).
[88] B. Mashhoon and U. Muench, “Length measurement in accelerated sys- tems,” Annalen Phys. 11, 532 (2002) [gr-qc/0206082].
[89] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and
W. K. Wootters, “Teleporting an unknown quantum state via dual clas- sical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[90] H. Buhrman, R. Cleve, Richard, S. Massar, and R. de Wolf, “Nonlocal- ity and communication complexity," Rev. Mod. Phys. 82, 665 (2010) [arXiv:0907.3584 [quant-ph]].
[91] R. Laiho, S. N. Molotkov and S. S. Nazin, “Teleportation of the relativis- tic quantum field,” Phys. Lett. A 275, 36 (2000) [quant-ph/0005067].
[92] V. Coffman, J. Kundu and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000) [quant-ph/9907047].
[93] M. Hotta, “Energy-Entanglement Relation for Quantum Energy Tele- portation,” Phys. Lett. A 374, 3416 (2010) [arXiv:1002.0200 [quant-ph]].
[94] M. Hotta, “Quantum Energy Teleportation: An Introductory Review,” arXiv:1101.3954 [quant-ph].
[95] M. Hotta, “Quantum measurement information as a key to energy extraction from local vacuums,” Phys. Rev. D 78, 045006 (2008) [arXiv:0803.2272 [physics.gen-ph]].
[96] M. Hotta, “Controlled Hawking Process by Quantum Energy Telepor- tation,” Phys. Rev. D 81, 044025 (2010) [arXiv:0907.1378 [gr-qc]].
[97] W. Pusz and S. L. Woronowicz, “Passive States and KMS States for General Quantum Systems,” Commun. Math. Phys. 58, 273 (1978).
[98] M. Banados, C. Teitelboim and J. Zanelli, “The Black hole in three- dimensional space-time,” Phys. Rev. Lett. 69, 1849 (1992) [hep- th/9204099].
[99] M. Hotta, “Quantum Energy Teleportation with Electromagnetic Field: Discrete vs. Continuous Variables,” Journal of Physics A: Mathematical and Theoretical, 43, 105305 (2010). arXiv:0908.2674 [quant-ph].
[100] T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, “EPR Pairs, Local Projections and Quantum Teleportation in Holography,” [arXiv:1604.01772 [hep-th]].
[101] M. A. Rajabpour, “Post measurement bipartite entanglement entropy in conformal field theories,” Phys. Rev. B 92, no. 7, 075108 (2015) [arXiv:1501.07831 [cond-mat.stat-mech]].
[102] M. Banados, “Three-dimensional quantum geometry and black holes,” AIP Conf. Proc. 484, 147 (1999) [hep-th/9901148].
[103] M. A. Rajabpour, “Entanglement entropy after a partial projective measurement in 1 + 1 dimensional conformal field theories: exact re- sults,” [arXiv:1512.03940 [hep-th]].
[104] J. L. Cardy, “Boundary conformal field theory,” [hep-th/0411189].
[105] P. Calabrese and J. L. Cardy, “Evolution of entanglement entropy in one-dimensional systems,” J. Stat. Mech. 0504, P04010 (2005) [cond- mat/0503393].
[106] P. Calabrese and J. L. Cardy, “Time-dependence of correlation func- tions following a quantum quench,” Phys. Rev. Lett. 96, 136801 (2006) [cond-mat/0601225].
[107] M. Nozaki, T. Numasawa and T. Takayanagi, “Holographic Lo- cal Quenches and Entanglement Density,” JHEP 1305, 080 (2013) [arXiv:1302.5703 [hep-th]].
[108] M. Miyaji, S. Ryu, T. Takayanagi and X. Wen, “Boundary States as Holographic Duals of Trivial Spacetimes,” JHEP 1505, 152 (2015) [arXiv:1412.6226 [hep-th]].
[109] X. Huang and F. L. Lin, “Entanglement renormalization and integral geometry,” JHEP 1512 (2015) 081 [arXiv:1507.04633 [hep-th]].
[110] D. Simmons-Duffin, “Projectors, Shadows, and Conformal Blocks,” JHEP 1404, 146 (2014) [arXiv:1204.3894 [hep-th]].
[111] B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, “A Stereo- scopic Look into the Bulk,” arXiv:1604.03110 [hep-th].
[112] B. Czech, D. Giataganas, F.-L. Lin and P.H. Liu, work in progress.
[113] B. Reznik, “Entanglement from the vacuum,” Found. Phys. 33, 167 (2003) [quant-ph/0212044].
[114] B. Reznik, A. Retzker and J. Silman, “Violating Bell’s inequalities in the vacuum,” Phys. Rev. A 71, no. 4, 042104 (2005) [quant-ph/0310058].
[115] J. Silman, B. Reznik, “Long-range entanglement in the Dirac vacuum", Phys. Rev. A 75, 052307 (2007) [quant-ph/0609212].
[116] G. Verdon-Akzam, E. Martin-Martinez and A. Kempf, “Asymptotically Limitless Quantum Energy Teleportation via Qudit Probes,” Phys. Rev. A 93, no. 2, 022308 (2016) [arXiv:1510.03751 [quant-ph]].
[117] M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams," J. Phys. A: Math. Gen. 27 (1994) L391-L398.
[118] Y. Aharonov, N. Erez, B. Reznik, “Superoscillations and tunneling times," Phys. Rev. A 65, 052124 (2002) [quant-ph/0110104].
[119] T. Faulkner, “The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT,” arXiv:1303.7221 [hep-th].
[120] T. Hartman, “Entanglement Entropy at Large Central Charge,” arXiv:1303.6955 [hep-th].
[121] M. Henningson and K. Skenderis, “The Holographic Weyl anomaly,” JHEP 9807, 023 (1998) [hep-th/9806087].
[122] V. Balasubramanian and P. Kraus, “A Stress tensor for Anti-de Sitter gravity,” Commun. Math. Phys. 208, 413 (1999) [hep-th/9902121].
[123] S. de Haro, S. N. Solodukhin and K. Skenderis, “Holographic recon- struction of space-time and renormalization in the AdS / CFT corre- spondence,” Commun. Math. Phys. 217, 595 (2001) [hep-th/0002230].
[124] J. D. Brown and J. W. York, Jr., “Quasilocal energy and conserved charges derived from the gravitational action,” Phys. Rev. D 47, 1407 (1993) [gr-qc/9209012].
[125] E. E. Flanagan, “Quantum inequalities in two-dimensional Minkowski space-time,” Phys. Rev. D 56, 4922 (1997) [gr-qc/9706006].
[126] R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer and A. C. Wall, “Proof of the Quantum Null Energy Condition,” Phys. Rev. D 93, no. 2, 024017 (2016) [arXiv:1509.02542 [hep-th]].
[127] J. Koeller and S. Leichenauer, “Holographic Proof of the Quantum Null Energy Condition,” arXiv:1512.06109 [hep-th].
[128] Wojciech H. Zurek, “Einselection and decoherence from an information theory perspective,” Annalen der Physik 9, 855864 (2000).
Harold Ollivier and Wojciech H. Zurek,“ Quantum Discord: A Mea- sure of the Quantumness of Correlations,” Phys. Rev. Lett. 88, 017901 (2001).
L. Henderson and V. Vedral,“Classical, quantum and total correlations,” Journal of Physics A 34, 6899 (2001).
[129] “Quantum discord for two-qubit X states,” Phys. Rev. A 81, 042105 (2010).