Toolbox for reconstructing quantum theory from rules on information acquisition

Philipp Andres Höhn

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5

We develop an operational approach for reconstructing the quantum theory of qubit systems from elementary rules on information acquisition. The focus lies on an observer $O$ interrogating a system $S$ with binary questions and $S$'s state is taken as $O$'s `catalogue of knowledge' about $S$. The mathematical tools of the framework are simple and we attempt to highlight all underlying assumptions. Four rules are imposed, asserting (1) a limit on the amount of information available to $O$; (2) the mere existence of complementary information; (3) $O$'s total amount of information to be preserved in-between interrogations; and, (4) $O$'s `catalogue of knowledge' to change continuously in time in-between interrogations and every consistent such evolution to be possible. This approach permits a {\it constructive} derivation of quantum theory, elucidating how the ensuing independence, complementarity and compatibility structure of $O$'s questions matches that of projective measurements in quantum theory, how entanglement and monogamy of entanglement, non-locality and, more generally, how the correlation structure of arbitrarily many qubits and rebits arises. The rules yield a reversible time evolution and a quadratic measure, quantifying $O$'s information about $S$. Finally, it is shown that the four rules admit two solutions for the simplest case of a single elementary system: the Bloch ball and disc as state spaces for a qubit and rebit, respectively, together with their symmetries as time evolution groups. The reconstruction for arbitrarily many qubits is completed in a companion paper [P. A. Höhn and C. S. P. Wever, Phys. Rev. A 95 (2017) 012102] where an additional rule eliminates the rebit case. This approach is inspired by (but does not rely on) the relational interpretation and yields a novel formulation of quantum theory in terms of questions.

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► References

[1] P. A. Höhn and C. S. P. Wever, ``Quantum theory from questions,'' Phys.Rev. A95 (2017) 012102, Preprint: arXiv:1511.01130.
https://doi.org/10.1103/PhysRevA.95.012102
arXiv:1511.01130

[2] P. A. Höhn and C. S. P. Wever, ``A reconstruction of real quantum theory from rules on information acquisition," to appear.

[3] R. D. Sorkin, ``On the Entropy of the Vacuum outside a Horizon,'' 10th Int. Conf. Gen. Rel. Grav. Contributed Papers II (1983) 734, arXiv:1402.3589.
arXiv:1402.3589

[4] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, ``A Quantum Source of Entropy for Black Holes,'' Phys.Rev. D34 (1986) 373-383.
https://doi.org/10.1103/PhysRevD.34.373

[5] T. Jacobson, ``Thermodynamics of space-time: The Einstein equation of state,'' Phys.Rev.Lett. 75 (1995) 1260-1263, arXiv:gr-qc/​9504004.
https://doi.org/10.1103/PhysRevLett.75.1260
arXiv:gr-qc/9504004

[6] T. Jacobson, ``Gravitation and vacuum entanglement entropy,'' Int.J.Mod.Phys. D21 (2012) 1242006, arXiv:1204.6349.
https://doi.org/10.1142/S0218271812420060
arXiv:1204.6349

[7] E. Bianchi and R. C. Myers, ``On the Architecture of Spacetime Geometry,'' Class.Quant.Grav. 31 no. 21, (2014) 214002, arXiv:1212.5183.
https://doi.org/10.1088/0264-9381/31/21/214002
arXiv:1212.5183

[8] E. T. Jaynes, ``Information theory and statistical mechanics," Phys. Rev. 106 (1957) 620.
https://doi.org/10.1103/PhysRev.106.620

[9] C. H. Bennett, ``The thermodynamics of computation-a review,'' Intern. J. Theor. Phys. 21, (1982) 905-940.
https://doi.org/10.1007/BF02084158

[10] K. Maruyama, F. Nori, and V. Vedral, ``Colloquium: The physics of maxwell's demon and information,'' Reviews of Modern Physics 81, (2009) 1.
https://doi.org/10.1103/RevModPhys.81.1

[11] S. Popescu, A. J. Short, and A. Winter, ``Entanglement and the foundations of statistical mechanics,'' Nature Physics 2, (2006) 754-758.
https://doi.org/10.1038/nphys444

[12] M. Horodecki and J. Oppenheim, ``Fundamental limitations for quantum and nanoscale thermodynamics,'' Nature communications 4 (2013).
https://doi.org/10.1038/ncomms3059

[13] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. Cambridge university press, 2010.

[14] L. Hardy, ``Quantum theory from five reasonable axioms,'' arXiv:quant-ph/​0101012 [quant-ph].
arXiv:quant-ph/0101012

[15] B. Dakic and C. Brukner, ``Quantum theory and beyond: Is entanglement special?,'' Deep Beauty: Understanding the Quantum World through Mathematical Innovation, Ed. H. Halvorson (Cambridge University Press, 2011) 365-392 (11, 2009), arXiv:0911.0695.
arXiv:0911.0695

[16] L. Masanes and M. P. Müller, ``A derivation of quantum theory from physical requirements,'' New Journal of Physics 13, (2011) 063001.
https://doi.org/10.1088/1367-2630/13/6/063001

[17] M. P. Müller and L. Masanes, ``Information-theoretic postulates for quantum theory,'' in Quantum Theory: Informational Foundations and Foils. Chiribella G., Spekkens R. (eds) Fund. Theories Phys., vol 181, Springer arXiv:1203.4516.
https://doi.org/10.1007/978-94-017-7303-4_5
arXiv:1203.4516

[18] L. Masanes, M. P. Müller, R. Augusiak, and D. Perez-Garcia, ``Existence of an information unit as a postulate of quantum theory,'' PNAS 110, 16373 (2013), arXiv:1208.0493.
https://doi.org/10.1073/pnas.1304884110
arXiv:1208.0493

[19] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Informational derivation of quantum theory,'' Physical Review A 84, (2011) 012311.
https://doi.org/10.1103/PhysRevA.84.012311

[20] G. de la Torre, L. Masanes, A. J. Short, and M. P. Müller, ``Deriving quantum theory from its local structure and reversibility,'' Physical Review Letters 109, (2012) 090403.
https://doi.org/10.1103/PhysRevLett.109.090403

[21] M. P. Müller and L. Masanes, ``Three-dimensionality of space and the quantum bit: how to derive both from information-theoretic postulates,'' New J. Phys. 15, 053040 (2013), arXiv:1206.0630.
https://doi.org/10.1088/1367-2630/15/5/053040
arXiv:1206.0630

[22] L. Hardy, ``Reconstructing quantum theory,'' arXiv:1303.1538.
arXiv:1303.1538

[23] H. Barnum, M. P. Müller, and C. Ududec, ``Higher-order interference and single-system postulates characterizing quantum theory,'' New J. Phys. 16 123029 (2014) arXiv:1403.4147.
https://doi.org/10.1088/1367-2630/16/12/123029
arXiv:1403.4147

[24] S. Kochen, ``A reconstruction of quantum mechanics,'' in Bertlmann R., Zeilinger A. (eds) Quantum [Un]Speakables II. (2017) The Frontiers Collection. Springer, Cham, arXiv:1306.3951.
https://doi.org/10.1007/978-3-319-38987-5_12
arXiv:1306.3951

[25] R. Oeckl, ``A local and operational framework for the foundations of physics ,'' arXiv:1610.09052.
arXiv:1610.09052

[26] J. B. Hartle, ``Quantum mechanics of individual systems,'' Am. J. of Phys. 36, (1968) 704-712.
https://doi.org/10.1119/1.1975096

[27] Q. Zheng and T. Kobayashi, ``Quantum optics as a relativistic theory of light,'' Physics Essays 9 (1996) 447-459.
https://doi.org/10.4006/1.3029255

[28] C. Rovelli, ``Relational quantum mechanics,'' Int.J.Theor.Phys. 35 (1996) 1637-1678, arXiv:quant-ph/​9609002.
https://doi.org/10.1007/BF02302261
arXiv:quant-ph/9609002

[29] M. Smerlak and C. Rovelli, ``Relational EPR,'' Found.Phys. 37 (2007) 427-445, arXiv:quant-ph/​0604064.
https://doi.org/10.1007/s10701-007-9105-0
arXiv:quant-ph/0604064

[30] A. Peres, Quantum theory: concepts and methods, vol. 57. Springer, 1995.

[31] A. Zeilinger, ``A foundational principle for quantum mechanics,'' Foundations of Physics 29, (1999) 631-643.
https://doi.org/10.1023/A:1018820410908

[32] C. Brukner and A. Zeilinger, ``Operationally invariant information in quantum measurements,'' Phys. Rev. Lett. 83 (1999) 3354-3357, arXiv:quant-ph/​0005084.
https://doi.org/10.1103/PhysRevLett.83.3354
arXiv:quant-ph/0005084

[33] C. Brukner, M. Zukowski, and A. Zeilinger, ``The essence of entanglement,'' arXiv:quant-ph/​0106119.
arXiv:quant-ph/0106119

[34] C. Brukner and A. Zeilinger, ``Information and fundamental elements of the structure of quantum theory,'' in ``Time, Quantum, Information'', eds. by L. Castell and O. Ischebeck (Springer, 2003), arXiv:quant-ph/​0212084.
arXiv:quant-ph/0212084

[35] C. Brukner and A. Zeilinger, ``Young's experiment and the finiteness of information,'' Phil. Trans. R. Soc. Lond. A 360 (2002) 1061, arXiv:quant-ph/​0201026.
https://doi.org/10.1098/rsta.2001.0981
arXiv:quant-ph/0201026

[36] C. A. Fuchs, ``Quantum mechanics as quantum information (and only a little more),'' arXiv:quant-ph/​0205039.
arXiv:quant-ph/0205039

[37] C. M. Caves and C. A. Fuchs, ``Quantum information: How much information in a state vector?,'' arXiv:quant-ph/​9601025.
arXiv:quant-ph/9601025

[38] C. M. Caves, C. A. Fuchs, and R. Schack, ``Quantum probabilities as bayesian probabilities,'' Phys. Rev. A 65 (2002) 022305, arXiv:quant-ph/​0106133.
https://doi.org/10.1103/PhysRevA.65.022305
arXiv:quant-ph/0106133

[39] C. M. Caves, C. A. Fuchs, and R. Schack, ``Unknown quantum states: the quantum de finetti representation,'' J. Math. Phys. 43, (2002) 4537-4559.
https://doi.org/10.1063/1.1494475

[40] R. W. Spekkens, ``Evidence for the epistemic view of quantum states: A toy theory,'' Physical Review A 75, (2007) 032110.
https://doi.org/10.1103/PhysRevA.75.032110

[41] R. W. Spekkens, ``Quasi-quantization: classical statistical theories with an epistemic restriction,'' in Quantum Theory: Informational Foundations and Foils. Chiribella G., Spekkens R. (eds) Fund. Theories Phys., vol 181, Springer arXiv:1409.5041.
https://doi.org/10.1007/978-94-017-7303-4_4
arXiv:1409.5041

[42] P. A. Höhn and M. P. Müller, ``An operational approach to spacetime symmetries: Lorentz transformations from quantum communication,'' New J. Phys. 18 (2016), 063026, arXiv:1412.8462.
https://doi.org/10.1088/1367-2630/18/6/063026
arXiv:1412.8462

[43] C. Rovelli, Quantum Gravity. Cambridge University Press, 2004.

[44] C. Rovelli, ``Time in quantum gravity: Physics beyond the Schrödinger regime,'' Phys.Rev. D43 (1991) 442-456.
https://doi.org/10.1103/PhysRevD.43.442

[45] C. Rovelli, ``What is observable in classical and quantum gravity?,'' Class.Quant.Grav. 8 (1991) 297-316.
https://doi.org/10.1088/0264-9381/8/2/011

[46] C. Rovelli, ``Quantum reference systems,'' Class.Quant.Grav. 8 (1991) 317-332.
https://doi.org/10.1088/0264-9381/8/2/012

[47] B. Dittrich, ``Partial and complete observables for canonical General Relativity,'' Class.Quant.Grav. 23 (2006) 6155-6184, arXiv:gr-qc/​0507106.
https://doi.org/10.1088/0264-9381/23/22/006
arXiv:gr-qc/0507106

[48] J. Tambornino, ``Relational observables in gravity: A review,'' SIGMA 8 (2012) 017, arXiv:1109.0740.
https://doi.org/10.3842/SIGMA.2012.017
arXiv:1109.0740

[49] M. Bojowald, P. A. Höhn, and A. Tsobanjan, ``Effective approach to the problem of time: general features and examples,'' Phys.Rev. D83 (2011) 125023, arXiv:1011.3040.
https://doi.org/10.1103/PhysRevD.83.125023
arXiv:1011.3040

[50] M. Bojowald, P. A. Höhn, and A. Tsobanjan, ``An Effective approach to the problem of time,'' Class.Quant.Grav. 28 (2011) 035006, arXiv:1009.5953.
https://doi.org/10.1088/0264-9381/28/3/035006
arXiv:1009.5953

[51] P. A. Höhn, E. Kubalova, and A. Tsobanjan, ``Effective relational dynamics of a nonintegrable cosmological model,'' Phys.Rev. D86 (2012) 065014, arXiv:1111.5193.
https://doi.org/10.1103/PhysRevD.86.065014
arXiv:1111.5193

[52] S. Popescu and D. Röhrlich, ``Quantum nonlocality as an axiom,'' Foundations of Physics 24, (1994) 379-385.
https://doi.org/10.1007/BF02058098

[53] M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski, ``Information causality as a physical principle,'' Nature 461, (2009) 1101-1104.
https://doi.org/10.1038/nature08400

[54] T. Paterek, B. Dakic, and C. Brukner, ``Theories of systems with limited information content,'' New J. Phys. 12 (2010) 053037, arXiv:0804.1423.
https://doi.org/10.1088/1367-2630/12/5/053037
arXiv:0804.1423

[55] J. Barrett, ``Information processing in generalized probabilistic theories,'' Physical Review A 75, (2007) 032304.
https://doi.org/10.1103/PhysRevA.75.032304

[56] L. Hardy, ``Foliable operational structures for general probabilistic theories,'' Deep Beauty: Understanding the Quantum World through Mathematical Innovation; Halvorson, H., Ed (2011) 409, arXiv:0912.4740.
arXiv:0912.4740

[57] L. Hardy, ``A formalism-local framework for general probabilistic theories, including quantum theory,'' Math. Struct. Comp. Science 23, (2013) 399, arXiv:1005.5164.
https://doi.org/10.1017/S0960129512000163
arXiv:1005.5164

[58] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Probabilistic theories with purification,'' Physical Review A 81, (2010) 062348.
https://doi.org/10.1103/PhysRevA.81.062348

[59] L. Masanes, M. P. Müller, D. Perez-Garcia, and R. Augusiak, ``Entanglement and the three-dimensionality of the bloch ball,'' J. Math. Phys. 55, 122203 (2014), arXiv:1111.4060.
https://doi.org/10.1063/1.4903510
arXiv:1111.4060

[60] C. Pfister and S. Wehner, ``An information-theoretic principle implies that any discrete physical theory is classical,'' Nature communications 4 (2013) 1851.
https://doi.org/10.1038/ncomms2821

[61] H. Barnum, J. Barrett, L. O. Clark, M. Leifer, R. Spekkens, N. Stepanik, A. Wilce, and R. Wilke, ``Entropy and information causality in general probabilistic theories,'' New J. Phys. 12 (2010) 033024, arXiv:0909.5075.
https://doi.org/10.1088/1367-2630/14/12/129401
arXiv:0909.5075

[62] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam, 1982.

[63] R. W. Spekkens, ``Contextuality for preparations, transformations, and unsharp measurements,'' Physical Review A 71, (2005) 052108.
https://doi.org/10.1103/PhysRevA.71.052108

[64] C. Caves, C. Fuchs, and R. Schack, ``Conditions for compatibility of quantum-state assignments,'' Phys. Rev. A 66 (Dec, 2002) 062111.
https://doi.org/10.1103/PhysRevA.66.062111

[65] N. D. Mermin, ``Compatibility of state assignments,'' J. Math. Phys. 43, (2002) 4560.
https://doi.org/10.1063/1.1495897

[66] C. A. Fuchs and A. Peres, ``Quantum state disturbance versus information gain: Uncertainty relations for quantum information,'' Phys.Rev. A53 (1996) 2038, arXiv:quant-ph/​9512023.
https://doi.org/10.1103/PhysRevA.53.2038
arXiv:quant-ph/9512023

[67] C. A. Fuchs, ``Information gain versus state disturbance in quantum theory,'' arXiv:quant-ph/​9611010.
arXiv:quant-ph/9611010

[68] E. Specker, ``Die logik nicht gleichzeitig entscheidbarer aussagen,'' Dialectica 14 (1960) 239.
https://doi.org/10.1111/j.1746-8361.1960.tb00422.x

[69] Y.-C. Liang, R. W. Spekkens, and H. M. Wiseman, ``Specker's parable of the overprotective seer: A road to contextuality, nonlocality and complementarity,'' Physics Reports 506, (2011) 1-39.
https://doi.org/10.1016/j.physrep.2011.05.001

[70] A. Cabello, ``Simple explanation of the quantum violation of a fundamental inequality,'' Physical review letters 110, (2013) 060402.
https://doi.org/10.1103/PhysRevLett.110.060402

[71] A. Cabello, ``Specker's fundamental principle of quantum mechanics,'' arXiv:1212.1756.
arXiv:1212.1756

[72] G. Chiribella and X. Yuan, ``Measurement sharpness cuts nonlocality and contextuality in every physical theory,'' arXiv:1404.3348.
arXiv:1404.3348

[73] Č. Brukner and A. Zeilinger, ``Information invariance and quantum probabilities,'' Foundations of Physics 39, (2009) 677-689.
https://doi.org/10.1007/s10701-009-9316-7

[74] S. Weinberg, ``Precision Tests of Quantum Mechanics,'' Phys.Rev.Lett. 62 (1989) 485.
https://doi.org/10.1103/PhysRevLett.62.485

[75] N. Gisin, ``Weinberg's non-linear quantum mechanics and supraluminal communications,'' Physics Letters A 143, (1990) 1-2.
https://doi.org/10.1016/0375-9601(90)90786-N

[76] J. Polchinski, ``Weinberg's nonlinear quantum mechanics and the EPR paradox,'' Phys.Rev.Lett. 66 (1991) 397-400.
https://doi.org/10.1103/PhysRevLett.66.397

[77] C. H. Bennett, D. Leung, G. Smith, and J. A. Smolin, ``Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems?,'' Phys.Rev.Lett. 103 (2009) 170502, arXiv:0908.3023.
https://doi.org/10.1103/PhysRevLett.103.170502
arXiv:0908.3023

[78] C. F. von Weizsäcker, The Structure of Physics. Springer-Verlag, Dordrecht, 2006.

[79] T. Görnitz and O. Ischebeck, An Introduction to Carl Friedrich von Weizsäcker's Program for a Reconstruction of Quantum Theory. in ``Time, Quantum, Information'', eds. by L. Castell and O. Ischebeck (Springer, 2003).
https://doi.org/10.1007/978-3-662-10557-3_17

[80] H. Lyre, ``Quantum theory of ur-objects as a theory of information,'' Int. J. Theor. Physics 34, (1995) 1541.
https://doi.org/10.1007/BF00676265

[81] B. Dakic and C. Brukner, ``The classical limit of a physical theory and the dimensionality of space,'' in Quantum Theory: Informational Foundations and Foils. Chiribella G., Spekkens R. (eds) Fund. Theories Phys., vol 181, Springer arXiv:1307.3984.
https://doi.org/10.1007/978-94-017-7303-4_8
arXiv:1307.3984

[82] N. Bohr, Atomic Theory and the Description of Nature. Cambridge University Press, 1961 (Reprint).

[83] C. M. Caves, C. A. Fuchs, and P. Rungta, ``Entanglement of formation of an arbitrary state of two rebits,'' Foundations of Physics Letters 14, (2001) 199.
https://doi.org/10.1023/A:1012215309321

[84] V. Coffman, J. Kundu, and W. K. Wootters, ``Distributed entanglement,'' Phys.Rev.A 61 (2000) 052306, arXiv:quant-ph/​9907047.
https://doi.org/10.1103/PhysRevA.61.052306
arXiv:quant-ph/9907047

[85] B. Regula, S. D. Martino, S. Lee, and G. Adesso, ``Strong monogamy conjecture for multiqubit entanglement: The four-qubit case,'' Phys. Rev. Lett. 113 (2014) 110501, arXiv:1405.3989.
https://doi.org/10.1103/PhysRevLett.113.110501
arXiv:1405.3989

[86] W. K. Wootters, ``Entanglement sharing in real-vector-space quantum theory,'' Foundations of Physics 42, (2012) 19.
https://doi.org/10.1007/s10701-010-9488-1

[87] W. K. Wootters, ``The rebit three-tangle and its relation to two-qubit entanglement,'' J. Phys. A: Math. Theor. 47 (2014) 424037.
https://doi.org/10.1088/1751-8113/47/42/424037

[88] A. Peres, ``Separability criterion for density matrices,'' Phys.Rev.Lett. 77 (1996) 1413-1415, arXiv:quant-ph/​9604005.
https://doi.org/10.1103/PhysRevLett.77.1413
arXiv:quant-ph/9604005

[89] R. Webster, Convexity. Oxford University Press, Oxford, 1994.

[90] S. Aaronson, ``Is quantum mechanics an island in theoryspace?,'' arXiv:quant-ph/​0401062.
arXiv:quant-ph/0401062

[91] C. Brukner and A. Zeilinger, ``Quantum measurement and shannon information, a reply to m. j. w. hall,'' arXiv:quant-ph/​0008091.
arXiv:quant-ph/0008091

[92] C. Brukner and A. Zeilinger, ``Conceptual inadequacy of the shannon information in quantum measurements,'' Phys.Rev.A 63 (2001) 022113, arXiv:quant-ph/​0006087.
https://doi.org/10.1103/PhysRevA.63.022113
arXiv:quant-ph/0006087

[93] A. J. P. Garner, M. P. Müller, and O. C. O. Dahlsten, ``The quantum bit from relativity of simultaneity on an interferometer,'' Proc. R. Soc. A473 (2017) 20170596, arXiv:1412.7112.
https://doi.org/10.1098/rspa.2017.0596
arXiv:1412.7112

[94] P. A. Höhn, ``Reflections on the information paradigm in quantum and gravitational physics,'' J. Phys. Conf. Ser. 880, 012014 (2017), arXiv:1706.06882.
https://doi.org/10.1088/1742-6596/880/1/012014
arXiv:1706.06882

[95] J. Hartle and S. Hawking, ``Wave function of the universe,'' Phys.Rev. D28 (1983) 2960-2975.
https://doi.org/10.1103/PhysRevD.28.2960

[96] M. Bojowald, ``Quantum cosmology,'' Lect.Notes Phys. 835 (2011) 1-308.
https://doi.org/10.1007/978-1-4419-8276-6

[97] A. Ashtekar and P. Singh, ``Loop Quantum Cosmology: A status report,'' Class.Quant.Grav. 28 (2011) 213001, arXiv:1108.0893.
https://doi.org/10.1088/0264-9381/28/21/213001
arXiv:1108.0893

[98] M. L. Dalla Chiara, ``Logical self reference, set theoretical paradoxes and the measurement problem in quantum mechanics,'' J. Philosophical Logic 6, (1977) 331.
https://doi.org/10.1007/BF00262066

[99] T. Breuer, ``The impossibility of accurate state self-measurements,'' Phil. Science (1995) 197.
https://doi.org/10.1086/289852

[100] L. Crane, ``Clock and category: Is quantum gravity algebraic?,'' J.Math.Phys. 36 (1995) 6180, arXiv:gr-qc/​9504038.
https://doi.org/10.1063/1.531240
arXiv:gr-qc/9504038

[101] F. Markopoulou, ``Quantum causal histories,'' Class.Quant.Grav. 17 (2000) 2059, arXiv:hep-th/​9904009.
https://doi.org/10.1088/0264-9381/17/10/302
arXiv:hep-th/9904009

[102] F. Markopoulou, ``Planck scale models of the universe,'' arXiv:gr-qc/​0210086.
arXiv:gr-qc/0210086

[103] F. Markopoulou, ``New directions in background independent quantum gravity,'' in Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. D. Oriti (2009), Cambridge University Press, 129 arXiv:gr-qc/​0703097.
arXiv:gr-qc/0703097

[104] L. F. Hackl and Y. Neiman, ``Horizon complementarity in elliptic de Sitter space,'' Phys. Rev. D 91 (2015), 044016, arXiv:1409.6753.
https://doi.org/10.1103/PhysRevD.91.044016
arXiv:1409.6753

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