Physics in Perspective

, Volume 16, Issue 1, pp 69–97 | Cite as

Informing Physics: Jacob Bekenstein and the Informational Turn in Theoretical Physics

  • Israel BelferEmail author


In his PhD dissertation in the early 1970s, the Mexican-Israeli theoretical physicist Jacob Bekenstein developed the thermodynamics of black holes using a generalized version of the second law of thermodynamics. This work made it possible for physicists to describe and analyze black holes using information-theoretical concepts. It also helped to transform information theory into a fundamental and foundational concept in theoretical physics. The story of Bekenstein’s work—which was initially opposed by many scientists, including Stephen Hawking—highlights the transformation within physics towards an information-oriented scientific mode of theorizing. This “informational turn” amounted to a mild-mannered revolution within physics, revolutionary without being rebellious.


Black holes thermodynamics black hole thermodynamics information theory information entropy generalized second law information bound quantum information theory Maxwell’s demon John Wheeler Jacob Bekenstein Stephen Hawking 



I would like to thank Professor Jacob Bekenstein for allowing me to meet with him and learn first-hand about his work in black hole thermodynamics, its development and ramifications. I would also like to thank Professor Silvan S. Schweber for the constant support and for suffering through the various drafts. This work was made possible thanks to the support of the Edelstein Center, Hebrew University, Jerusalem.


  1. 1.
    The name of the gravitational singularity was popularized by John A. Wheeler. According to Bekenstein, Of Gravity, Black Holes and Information (Rome: Di Renzo Editore, 2006), 24, in a lecture before a large audience. Wheeler was looking for a shorthand version of “completely gravitationally collapsed object,” and picked up the name as a suggestion from “a voice in the audience.” Wheeler gave an account of the first printed use of “black hole” in the Proceedings of the American Association for the Advancement of Science (AAAS) in New York, speaking in front of the Society of Sigma Xi; John Archibald Wheeler, Geons, Black Holes and Quantum Foam: A Life in Physics (New York: Norton, 1999), 296. Tired of using the long aforementioned phrase, Wheeler took up the suggested “black hole” from the audience. The term was reported even earlier by Ann Ewing in “‘Black Holes’ in Space,” Science News Letter, January 18, 1964, 39. The combination “black hole” was famously used in reference to “The Black Hole of Calcutta,” a dungeon in Fort William, where numerous English and Anglo-Indian soldiers and civilians were held and many died. Leonard Susskind, The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics (New York: Little, Brown, 2008), 288–289, describes meeting a fellow “Black Hole Expert” in an English bar, who turned out to be speaking not of his specialty but of the one in India. Also cf. Tom Siegfried, “50 Years Later, It’s Hard to Say Who Named Black Holes,” Science News Letter, December 23, 2013,, accessed January 2, 2014.
  2. 2.
    The Bekensteins moved to Mexico with other families escaping Nazi oppression to a country with relatively lax immigration laws and possibilities of prolonged temporary visits; cf. Adina Cimet, Ashkenazi Jews in Mexico: Ideologies in the Structuring of a Community (Albany, NY: SUNY Press, 1997), 13, 22, 112. The Bekensteins and many other families found a temporary haven there, though their final destination was New York. This was reached, however, with great effort. Jacob’s father made his way to the US first illegally—and was even arrested—eventually securing passage for the rest of the family.Google Scholar
  3. 3.
    The Unified Honors Program (still in place) condensed the required courses in a way that forced Bekenstein to take the second part of the electromagnetic theory course, given by David Stoler, before the first. This made his first acquaintance with Green’s functions very challenging.Google Scholar
  4. 4.
    This shift has a linear and a quadratic element, first derived for the hydrogen atom by Paul Epstein and Karl Schwarzschild. Bekenstein was able to derive the fourth order term. Jacob D. Bekenstein and Joseph B. Krieger, “Stark Effect in Hydrogenic Atoms: Comparison of Fourth-Order Perturbation Theory with WKB Approximation,” Physical Review 188 (1969), 130–9; “Stark Effect in Hydrogen Atoms for Nonuniform Fields,” Journal of Mathematical Physics 11 (1970), 2721–7.Google Scholar
  5. 5.
    Bekenstein, Of Gravity (ref. 1), 10.Google Scholar
  6. 6.
    Bekenstein spent the summer of 1969 as an assistant to Dino Goulianos and Dave Bartlett, two physicists at Princeton. He examined the velocities of neutrons stripped from deuterium nuclei (deuterons) in order to examine the deuteron quantum state proposed by Swedish physicist Lamek Hulthén (the Hulthén potential). See Lamek Hulthén and Masao Sugawara, “The Two-Nucleon Problem,” Handbuch der Physik 39 (1957), 1–143.Google Scholar
  7. 7.
    Bekenstein, Of Gravity (ref. 1) 17.Google Scholar
  8. 8.
    Brillouin saw information theory as expanding the statistical methods used in experimental data analysis. Jaynes went further, adopting the subjective aspect of information entropy, his principle of maximum entropy according to which the scientist must assume and quantify the level of ignorance regarding the system. Léon Brillouin, Science and Information Theory (Waltham, MA: Academic Press, 1956); Edwin T. Jaynes, “Information Theory and Statistical Mechanics,” Physical Review 106 (1957), 620–30; “Information Theory and Statistical Mechanics. II,” Physical Review 108 (1957), 171–90.Google Scholar
  9. 9.
    Terry M Christensen, “John Archibald Wheeler: A Study of Mentoring in Modern Physics,” PhD diss., Oregon State University, 2009, 139.Google Scholar
  10. 10.
    Ibid., 272.Google Scholar
  11. 11.
    Wheeler hosted a brown-bag lunch with light discussion on current topics (using one of the bags to signal the end to a discussion).Google Scholar
  12. 12.
    Wheeler, Geons, Black Holes and Quantum Foam (ref. 1), 63–64.Google Scholar
  13. 13.
    Roger Penrose, “Gravitational Collapse: The Role of General Relativity,” Nuovo Cimento Rivista 1 (1969), 252.Google Scholar
  14. 14.
    Demetrios Christodoulou, “Reversible and Irreversible Transformations in Black-Hole Physics,” Physical Review Letters 25 (1970), 1596–7; Demetrios Christodoulou and Remo Ruffini, “Reversible Transformations of a Charged Black Hole,” Physical Review D 4 (1971), 3552–5; Penrose, “Gravitational Collapse” (ref. 13); R. Penrose and R. M. Floyd, “Extraction of Rotational Energy from a Black Hole,” Nature 229 (1971), 177–9.Google Scholar
  15. 15.
    Jacob D. Bekenstein, “Energy Cost of Information Transfer,” Physical Review Letters 46 (1981), 623–6; “Universal Upper Bound on the Entropy-to-Energy Ratio for Bounded Systems,” Physical Review D 23 (1981), 287–98; “Entropy Content and Information Flow in Systems with Limited Energy,” Physical Review D 30 (1984), 1669–79; “The Limits of Information,” Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 32 (2001), 511–24.Google Scholar
  16. 16.
    Jacob D. Bekenstein, “Baryon Number, Entropy, and Black Hole Physics,” PhD diss., Princeton University, 1972, 62–66.Google Scholar
  17. 17.
    Bekenstein diagrams this by depicting the familiar gravitational field-lines emanating from the particle getting warped by the singularity as the particle approaches it. Once the particle crosses the event-horizon, even this warped version of the field is suddenly annulled, the lines unite with the singularity’s center, and this fundamental physical attribute of matter is apparently annihilated or transcended.Google Scholar
  18. 18.
    “A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content concerning which I am convinced that, within the framework of applicability of its basic concepts, it will never be overthrown (for the special attention of those who are skeptics on principle).” Albert Einstein, “Autobiographical Notes,” in P. A. Schilpp, ed., Albert Einstein: Philosopher-Scientist (Chicago: Open Court Publishing, 1979), 33.Google Scholar
  19. 19.
    Bekenstein attributes the name and the idea of this theorem to Wheeler; Bekenstein, Of Gravity, Black Holes and Information (ref. 1) 30–31. Wheeler tells the story a little differently, with Bekenstein coming up with the idea and himself coining the “No-Hair Theorem,” which Feynman, his older student, found in poor taste. John Archibald Wheeler, “Feynman and Jacob Bekenstein,” accessed June 21, 2012,
  20. 20.
    “The Wheeler demon hides information in a black hole. More precisely, it throws there the entropy which signaled some process whereby information in a physical system was lost through mixing with environment. Effectively it hides information. Of course nowadays when we know that the black hole will eventually evaporate we can think of the whole process as an erasure.” J. Bekenstein to author, July 15, 2011.Google Scholar
  21. 21.
    Claude Elwood Shannon, A Mathematical Theory of Communication (Bell System technical journal, 1948). Shannon’s definition of information as a measure in binary digits or bits is the basis for modern communication and computation technologies. Multiple versions of information measures have branched out from there, and relate to the branch of science where they are applied. See Tom Siegfried, The Bit and the Pendulum: From Quantum Computing to M TheoryThe New Physics of Information (New York: Wiley, 2000), 175–176, 200. Information as algorithmic depth (number of steps to achieve a state), coding capacity (shortest length of generating program), and statistical degrees of freedom, are connected to investigations into complexity, thermodynamics, and quantum Information—with differing goals. The counterintuitive basic measure of information-theoretical analysis is the measure of missing data or negentropy; see Léon Brillouin, Science and Information Theory, (Waltham, MA: Academic Press, 1956), 153–6.Google Scholar
  22. 22.
    Bekenstein, “Baryon Number, Entropy, and Black Hole Physics” (ref. 16), 109, 178; Leon Brillouin, “Maxwell’s Demon Cannot Operate: Information and Entropy. I,” Journal of Applied Physics 22 (1951), 334–7.Google Scholar
  23. 23.
    Bekenstein, “Baryon Number, Entropy, and Black Hole Physics” (ref. 16), 108–14 (“Of Demons, Black Holes and Entropy”).Google Scholar
  24. 24.
    Ibid., 107.Google Scholar
  25. 25.
    Jacob D. Bekenstein, interview by Israel Belfer, September 8, 2009.Google Scholar
  26. 26.
    Jacob D. Bekenstein, “The Limits of Information,” Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 32 (2001), 511–24.Google Scholar
  27. 27.
    A rotating (Kerr) black hole has another relevant surface besides the spherical event horizon. This is the oblate spheroid called the “ergosphere” (for the Greek ἕργον, meaning work). Penrose proposed that work could be extracted from the singularity. Within the ergosphere, strong Lenz-Thiring frame-dragging occurs: spacetime is dragged in the rotational direction of the black hole, i.e., faster than the local speed of light. The Penrose process is the splitting of an object in the ergosphere so that one part manages to escape the gravity well, with greater mass-energy than the whole object had before the frame-dragging influenced it. Thus energy is taken away from the black hole. This splitting maneuver is a precursor to Hawking’s later discovery of black-hole evaporation through particle pair production and uncompleted annihilation close to the event horizon, which was discarded at the time for various reasons; see Stephen. W. Hawking, “Black Hole Explosions?” Nature 248 (1974), 30–1.Google Scholar
  28. 28.
    Remo Ruffini and John A. Wheeler, “Introducing the Black Hole,” Physics Today 24(1) (1971), 30–41.Google Scholar
  29. 29.
    Penrose and Floyd, “Extraction of Rotational Energy from a Black Hole” (ref. 14).Google Scholar
  30. 30.
    S. W. Hawking and R. Penrose, “The Singularities of Gravitational Collapse and Cosmology,” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 314 (1970), 529–48; S. W. Hawking, “Gravitational Radiation from Colliding Black Holes,” Physical Review Letters 26 (1971), 1344–6.Google Scholar
  31. 31.
    Bekenstein, “The Limits of Information” (ref. 26).Google Scholar
  32. 32.
    The mathematical homology of information-entropy and that of thermodynamics is a classic example of Steiner’s modern scientific Pythagorean mode; that is, a mathematical analogy connecting older theories with new directions of modeling, with the analogy sometimes itself serving as enough of a basis for developing a physical model (isospin, for example). Steiner calls this strategy “Pythagorean” or “formalist,” depending on the centrality of mathematical syntax involved. Cf. Mark Steiner, The Applicability of Mathematics As a Philosophical Problem (Cambridge, MA: Harvard University Press, 1998).Google Scholar
  33. 33.
    Bekenstein, “The Limits of Information” (ref. 26), 515.Google Scholar
  34. 34.
    Ibid.; Bekenstein, “Baryon Number, Entropy, and Black Hole Physics” (ref. 16), 129–30.Google Scholar
  35. 35.
    Bekenstein, “The Limits of Information” (ref. 26), 110. For the classic work connecting information and physics through the demon’s sorting of molecules, see Leó Szilárd, “Über die Entropieverminderung in einem thermodynamischen System bei eingriffen intelligenter Wesen,” Zeitschrift für Physik 53 (1929), 840–856. Szilard conjectured that the entropy cost of sorting lies in acquiring the information. Later, Bennett showed that the process of manipulating information is reversible and that the core of the entropy cost was—counterintuitively—in the erasure: Charles H. Bennett, “Logical Reversibility of Computation,” IBM Journal of Research and Development 17 (1973), 525–32.Google Scholar
  36. 36.
    Bekenstein, “The Limits of Information” (ref. 26), 110–1; Harvey Leff and Andrew F. Rex, eds., Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing (Oxford: Taylor & Francis, 2002), 388–389.Google Scholar
  37. 37.
    Bekenstein, “Baryon Number, Entropy, and Black Hole Physics” (ref. 16), 111–26.Google Scholar
  38. 38.
    Jacob D. Bekenstein, “Do We Understand Black Hole Entropy?” ArXiv Preprint Gr-qc/9409015 (1994) 39–58.Google Scholar
  39. 39.
    Jacob D. Bekenstein, “Generalized Second Law of Thermodynamics in Black-Hole Physics,” Physical Review D 9 (1974), 3292–300. If P is the probability of a given alphabetical occurrence that is not foreseeable (expressed mathematically as a binary configuration, a register of bits x i), Shannon defined the information entropy as H(x) = – i P(x i ) ln (P(x i )); by comparison, Boltzmann defined entropy as S = –k ∑ i P i ln (P i ) where P i represents the possible physical configurations and k is Boltzmann’s constant. In later expositions of black hole entropy, this homology is more pronounced than in Bekenstein’s dissertation, when such a connection via the informational context was not yet intuitive for other physicists.Google Scholar
  40. 40.
  41. 41.
    Jacob D. Bekenstein, “Black Holes and Entropy,” Physical Review D 7 (1973), 2333–2346; Bekenstein, “Generalized Second Law of Thermodynamics in Black-Hole Physics” (ref. 39).Google Scholar
  42. 42.
    That is, S 0 + S bh ≥ 0. Jacob D. Bekenstein, “Black-Hole Thermodynamics,” Physics Today 33(1) (1980), 24–31.Google Scholar
  43. 43.
    Bekenstein, “Baryon Number, Entropy, and Black Hole Physics” (ref. 16), 137, “An Alternative Approach to Black Hole Entropy.” Bekenstein attributes the formulation of the equation (136, eq. 10.8) where Planck’s length is the denominator to Wheeler, who spoke of using Planck’s length in conversation with Bekenstein on April 2, 1971.Google Scholar
  44. 44.
    It is also additive (with respect to A), as is the logarithmic form of the Gibbs-Bolzmann entropy function. See Jacob D. Bekenstein, “Bekenstein-Hawking Entropy,” Scholarpedia, 2008,, figure 1.
  45. 45.
    Bekenstein, “Black Holes and Entropy” (ref. 41), 2335.Google Scholar
  46. 46.
    Chapter II (“Is Baryon number defined for black holes?”), III (“A static black hole can possess no exterior meson fields”), and IV (“A stationary black hole can possess no exterior meson fields”) of the dissertation were published in article form as J. D. Bekenstein, “Transcendence of the Law of Baryon-Number Conservation in Black-Hole Physics,” Physical Review Letters 28 (1972), 452–5.Google Scholar
  47. 47.
    Bekenstein, “Black Holes and Entropy” (ref. 41), 2336.Google Scholar
  48. 48.
    Chapter IX of the dissertation: Bekenstein, “Baryon Number, Entropy, and Black Hole Physics” (ref. 16), 103–126. In this chapter, the parallels between black holes and thermodynamics are introduced: “The irreversibility of an increase in the area of a black hole is reminiscent of the irreversibility of an increase in the entropy of a closed thermodynamical system. Similarly, the increase in horizon area when black holes merge parallels the increase in total entropy when thermodynamics systems are combined. The results of Christodoulou and Hawking thus play the role (at least formally) of the second law in black hole physics” (104).Google Scholar
  49. 49.
    Bekenstein, “Generalized Second Law of Thermodynamics in Black-Hole Physics” (ref. 39), 3297.Google Scholar
  50. 50.
    Bekenstein, “Black Holes and Entropy” (ref. 41), 2339.Google Scholar
  51. 51.
    Ibid., 2338.Google Scholar
  52. 52.
    The Austrian sociologist Karin Knorr-Cetina has noted of experimental particle physics that “it is quite remarkable how much one can do by mobilizing negative knowledge” scientifically: Karin Knorr-Cetina, Epistemic Cultures: How the Sciences Make Knowledge (Cambridge, MA: Harvard University Press, 1999), 65.Google Scholar
  53. 53.
    Bekenstein, Of Gravity, Black Holes and Information (ref. 1), 60.Google Scholar
  54. 54.
    Jacob D. Bekenstein, “Black Holes and the Second Law,” Lettere al Nuovo Cimento 4 (1972), 737–40; “Black Holes and Entropy,” Physical Review D 7 (1973), 2333–46.Google Scholar
  55. 55.
    Bekenstein, “Universal Upper Bound on the Entropy-to-Energy Ratio” (ref. 15); “The Limits of Information” Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 32 (2001), 511–24; “How Does the Entropy/Information Bound Work?” Foundations of Physics 35 (2005), 1805–23.Google Scholar
  56. 56.
    Bekenstein, “Black-Hole Thermodynamics” (ref. 42); see also ref. 55.Google Scholar
  57. 57.
    James M Bardeen, Brandon Carter, and Stephen W. Hawking, “The Four Laws of Black Hole Mechanics,” Communications in Mathematical Physics (1965-1997) 31 (1973), 161–170.Google Scholar
  58. 58.
    In Bekenstein’s case, this included an initial rejection by Hawking; see Dennis Overbye, Lonely Hearts of the Cosmos: The Scientific Quest for the Secret of the Universe (New York: HarperCollins, 1991), 107, 112; Michael White and John R. Gribbin, Stephen Hawking: A Life in Science (Washington, DC: Joseph Henry Press, 2002), 124–5. For Hawking’s acceptance, see Stephen W. Hawking, A Brief History of Time: From the Big Bang to Black Holes (New York: Bantam Books, 1988), 99. For the trivialization of Bekenstein’s “basically correct” idea, see Stephen W. Hawking, Black Holes And Baby Universes And Other Essays (New York: Bantam Books, 1994), 105.Google Scholar
  59. 59.
    Bardeen, Carter, and Hawking, “The Four Laws of Black Hole Mechanics” (ref. 57).Google Scholar
  60. 60.
    Ibid., 162.Google Scholar
  61. 61.
    Kristine Larsen, Stephen Hawking: A Biography (Westport, CT: Greenwood Publishing Group, 2005), 38.Google Scholar
  62. 62.
    The four laws: I. (corresponding to the usual second law) The area A of the event horizon of each black hole does not decrease with time. II. Relates the surface gravity of the black hole to a “temperature”, emphasizing that it is in fact thermally zero. III. (corresponding to the 0th law) The surface gravity is constant over the event horizon. IV (corresponding to the usual third law) It is impossible (however not rigorously proven) by any procedure, no matter how idealized (close to reversible), to reduce the surface gravity to zero by a finite sequence of operations. Cf. Bardeen, Carter, and Hawking, “The Four Laws of Black Hole Mechanics” (ref. 57), 167–70.Google Scholar
  63. 63.
    Kip S. Thorne, Black Holes and Time Warps: Einstein’s Outrageous Legacy (New York: Norton, 1994), 427.Google Scholar
  64. 64.
    Bardeen, Carter, and Hawking, “The Four Laws of Black Hole Mechanics” (ref. 58), 162.Google Scholar
  65. 65.
    Bekenstein, Of Gravity, Black Holes and Information (ref. 1), 27–43.Google Scholar
  66. 66.
    Hawking, “Black Hole Explosions?” (ref. 27).Google Scholar
  67. 67.
    Starobinsky was the student of eminent Russian physicist Yakov Zeldovich (in their own effort to combine quantum mechanics and general relativity). This was in August and September 1973 in Warsaw, at a celebration of the 500th anniversary Polish astronomer Nicolas Copernicus’ birth. See A. A. Starobinsky, “Amplification of Waves Reflected from Kerr Black Holes,” Gravitational Radiation and Gravitational Collapse 64 (1974), 94.Google Scholar
  68. 68.
    Larsen, Stephen Hawking (ref. 61), 40.Google Scholar
  69. 69.
    Hawking, “Black Hole Explosions?” (ref. 27), 30.Google Scholar
  70. 70.
    Hawking, A Brief History of Time (ref. 58) 108.Google Scholar
  71. 71.
    The now-famous 1997 “black-hole bet” concerning the information-content and information-retainability of black holes concerned the loss of information of a particle entering a black hole: Is the quantum state lost forever (thermally irreversibly mixed) or is there a lasting effect, a persistence of information in some form? Stephen Hawking and Kip Thorne thought a full annihilation of information occurs, and John Preskil claimed a mechanism that retrieves the information. See Kip Thorne, John P. Bentley, and Stephen W. Hawking, “Black Hole Information Bet,” 02 1997, The resolution of the bet is itself a matter of indecision, keeping the fate of information ultimately undetermined. Hawking declared a loss in the bet (at the 17th International Conference on General Relativity and Gravitation, on July 21, 2004 in Dublin) in his description of black hole radiation brought on by pair-creation on the event horizon of a black hole, separating the pair and preventing the self-annihilation.
  72. 72.
    Susskind, The Black Hole War (ref. 1), 438.Google Scholar
  73. 73.
    This happened in the mansion of physics enthusiast and motivational seminar guru Jack Rosenberg, where the EST (Erhard Seminars Training, named after his lucrative self-help programs) conference/workshop was held in San Francisco, 1983. The participants included such outstanding theorists as Frank Wilczek, Murray Gell-Mann, Sheldon Glashow, Dave Finkelstein, Savas Dimopoulos, Gerard ‘t Hooft, Stephen Hawking, and Leonard Susskind. Hawking, through his translator (due to his then unassisted but deteriorating speech) Martin Rocek, conveyed the prospect of total information annihilation (ibid., 17–20).Google Scholar
  74. 74.
    Ibid., title page.Google Scholar
  75. 75.
    Leonard Susskind and James Lindesay, An Introduction To Black Holes, Information And The String Theory Revolution: The Holographic Universe (Singapore: World Scientific, 2005), viii.Google Scholar
  76. 76.
    Bekenstein, “The Limits of Information” (ref. 55).Google Scholar
  77. 77.
    Jacob D. Bekenstein, “Black Holes and Information Theory,” Contemporary Physics 45 (2004), 31–43.Google Scholar
  78. 78.
    Gerard’t Hooft, “Dimensional Reduction in Quantum Gravity,” in Salamfestschrift, A Collection of Talks from the Conference on Highlights of Particle and Condensed Matter Physics (World Scientific Publishing Company, 1993), vol. 4; arXiv:gr-qc/9310026.Google Scholar
  79. 79.
    Bekenstein, “How Does the Entropy/Information Bound Work?” (ref. 55).Google Scholar
  80. 80.
    Here, reversibility was uniquely interpreted by Christodoulou. See Bekenstein, “Baryon Number, Entropy, and Black Hole Physics” (ref. 16), chapter XI; Shahar Hod, “Best Approximation to a Reversible Process in Black-Hole Physics and the Area Spectrum of Spherical Black Holes,” Physical Review D 59 (1998), 024014.Google Scholar
  81. 81.
    Bekenstein, Of Gravity, Black Holes and Information (ref. 1), 109.Google Scholar
  82. 82.
    Ibid., 111.Google Scholar
  83. 83.
    John Archibald Wheeler and Wojciech Hubert Żurek, Quantum Theory and Measurement (Princeton University Press, 1983), 209.Google Scholar
  84. 84.
    John Archibald Wheeler, At Home in the Universe (College Park, MD: American Institute of Physics, 1994), 299.Google Scholar
  85. 85.
    The results achieved contradicted the accepted wisdom on the subject found in 1962 by D. Lebedev and L. Levitin was rejected by Physical Review Letters. A second version was published in the Physical Review: Jacob D. Bekenstein, “Communication and Energy,” Physical Review A 37 (1988), 3437–49.Google Scholar
  86. 86.
    Two complementary research agendas connect the many disciplines involved in quantum information: 1) the applicability of information and computation theories (transmission, compression, cryptography, quantum computation) in a quantum setting; 2) the formulation of quantum mechanics in information theoretical terms, with such no-go theorems (such as the no-cloning theorem, the no deleting theorem) and bounds such as the Holevo bound on the amount of information that can be known about a quantum system. See Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information (Cambridge: Cambridge University Press, 2000).Google Scholar
  87. 87.
    Susskind, The Black Hole War (ref. 1), 9, 179.Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.The Jacques Loeb Centre for the History and Philosophy of the Life SciencesBen-Gurion University of the NegevBeer ShevaIsrael

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