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

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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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[1] P. A. Höhn and C. S. P. Wever, ``Quantum theory from questions,'' Phys.Rev. A95 (2017) 012102, Preprint: 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.

[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.

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

[6] T. Jacobson, ``Gravitation and vacuum entanglement entropy,'' Int.J.Mod.Phys. D21 (2012) 1242006, 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.

[8] E. T. Jaynes, ``Information theory and statistical mechanics," Phys. Rev. 106 (1957) 620.

[9] C. H. Bennett, ``The thermodynamics of computation-a review,'' Intern. J. Theor. Phys. 21, (1982) 905-940.

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

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

[12] M. Horodecki and J. Oppenheim, ``Fundamental limitations for quantum and nanoscale thermodynamics,'' Nature communications 4 (2013).

[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].

[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.

[16] L. Masanes and M. P. Müller, ``A derivation of quantum theory from physical requirements,'' New Journal of Physics 13, (2011) 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.

[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.

[19] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Informational derivation of quantum theory,'' Physical Review A 84, (2011) 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.

[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.

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

[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.

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

[26] J. B. Hartle, ``Quantum mechanics of individual systems,'' Am. J. of Phys. 36, (1968) 704-712.

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

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

[29] M. Smerlak and C. Rovelli, ``Relational EPR,'' Found.Phys. 37 (2007) 427-445, 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.

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

[33] C. Brukner, M. Zukowski, and A. Zeilinger, ``The essence of entanglement,'' 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.

[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.

[36] C. A. Fuchs, ``Quantum mechanics as quantum information (and only a little more),'' 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.

[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.

[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.

[40] R. W. Spekkens, ``Evidence for the epistemic view of quantum states: A toy theory,'' Physical Review A 75, (2007) 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.

[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.

[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.

[45] C. Rovelli, ``What is observable in classical and quantum gravity?,'' Class.Quant.Grav. 8 (1991) 297-316.

[46] C. Rovelli, ``Quantum reference systems,'' Class.Quant.Grav. 8 (1991) 317-332.

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

[48] J. Tambornino, ``Relational observables in gravity: A review,'' SIGMA 8 (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.

[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.

[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.

[52] S. Popescu and D. Röhrlich, ``Quantum nonlocality as an axiom,'' Foundations of Physics 24, (1994) 379-385.

[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.

[54] T. Paterek, B. Dakic, and C. Brukner, ``Theories of systems with limited information content,'' New J. Phys. 12 (2010) 053037, arXiv:0804.1423.

[55] J. Barrett, ``Information processing in generalized probabilistic theories,'' Physical Review A 75, (2007) 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.

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

[58] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Probabilistic theories with purification,'' Physical Review A 81, (2010) 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.

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

[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.

[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.

[64] C. Caves, C. Fuchs, and R. Schack, ``Conditions for compatibility of quantum-state assignments,'' Phys. Rev. A 66 (Dec, 2002) 062111.

[65] N. D. Mermin, ``Compatibility of state assignments,'' J. Math. Phys. 43, (2002) 4560.

[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.

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

[68] E. Specker, ``Die logik nicht gleichzeitig entscheidbarer aussagen,'' Dialectica 14 (1960) 239.

[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.

[70] A. Cabello, ``Simple explanation of the quantum violation of a fundamental inequality,'' Physical review letters 110, (2013) 060402.

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

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

[73] Č. Brukner and A. Zeilinger, ``Information invariance and quantum probabilities,'' Foundations of Physics 39, (2009) 677-689.

[74] S. Weinberg, ``Precision Tests of Quantum Mechanics,'' Phys.Rev.Lett. 62 (1989) 485.

[75] N. Gisin, ``Weinberg's non-linear quantum mechanics and supraluminal communications,'' Physics Letters A 143, (1990) 1-2.

[76] J. Polchinski, ``Weinberg's nonlinear quantum mechanics and the EPR paradox,'' Phys.Rev.Lett. 66 (1991) 397-400.

[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.

[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).

[80] H. Lyre, ``Quantum theory of ur-objects as a theory of information,'' Int. J. Theor. Physics 34, (1995) 1541.

[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.

[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.

[84] V. Coffman, J. Kundu, and W. K. Wootters, ``Distributed entanglement,'' Phys.Rev.A 61 (2000) 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.

[86] W. K. Wootters, ``Entanglement sharing in real-vector-space quantum theory,'' Foundations of Physics 42, (2012) 19.

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

[88] A. Peres, ``Separability criterion for density matrices,'' Phys.Rev.Lett. 77 (1996) 1413-1415, 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.

[91] C. Brukner and A. Zeilinger, ``Quantum measurement and shannon information, a reply to m. j. w. hall,'' 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.

[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.

[94] P. A. Höhn, ``Reflections on the information paradigm in quantum and gravitational physics,'' J. Phys. Conf. Ser. 880, 012014 (2017), arXiv:1706.06882.

[95] J. Hartle and S. Hawking, ``Wave function of the universe,'' Phys.Rev. D28 (1983) 2960-2975.

[96] M. Bojowald, ``Quantum cosmology,'' Lect.Notes Phys. 835 (2011) 1-308.

[97] A. Ashtekar and P. Singh, ``Loop Quantum Cosmology: A status report,'' Class.Quant.Grav. 28 (2011) 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.

[99] T. Breuer, ``The impossibility of accurate state self-measurements,'' Phil. Science (1995) 197.

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

[101] F. Markopoulou, ``Quantum causal histories,'' Class.Quant.Grav. 17 (2000) 2059, arXiv:hep-th/​9904009.

[102] F. Markopoulou, ``Planck scale models of the universe,'' 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.

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

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