Abstract
Landauer’s principle claims that “Information is Physical”. It is not surprising that its conceptual antithesis, Wheeler’s “It from Bit”, has been more popular among computer scientists — in the form of the Church-Turing hypothesis: All natural processes can be computed by a universal Turing machine; physical laws then become descriptions of subsets of observable, as opposed to merely possible, computations. Switching back and forth between the two traditional styles of thought, motivated by quantum-physical Bell correlations and the doubts they raise about fundamental space-time causality, we look for an intrinsic, physical randomness notion and find one around the second law of thermodynamics. Bell correlations combined with complexity as randomness tell us that beyond-Turing computations are either physically impossible, or they can be carried out by “devices” as simple as individual photons.
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Notes
- 1.
It has been shown [4] that if causality is dropped but logical consistency maintained, then a rich world opens — comparable to the one between locality and signaling.
- 2.
Note furthermore that the definition is consistent with full determinism: A random variable with trivial distribution is independent of every other (even itself).
- 3.
See, e.g., Allen, W., Husbands and Wives (1992): The protagonist Sally is explaining why her marriage did not work out. First she does not know, then she realizes: “It’s the second law of thermodynamics: sooner or later everything turns to shit. That’s my phrasing, not the Encyclopedia Britannica”.
- 4.
Boltzmann imagined further that the universe had started in a completely “uniform” state, so the entire, rich reality perceived would be a simple fluctuation. (Note that the fact that this fluctuation is extremely unlikely is irrelevant if we can condition on our existence, given our discussing this.) He may have been aware that this way of thinking leads straight into solipsism: “My existence alone, simply imagining my environment, seems much more likely than the actual existence of all people around me, let alone all the visible galaxies, etc.” — he killed himself in a hotel room in Duino, Italy; it has been told that this was also related to “mobbing” by Mach in Vienna. In any case, we choose to comfort us today with the somewhat religious assumption that the universe initiated in a low-entropy state, called the big bang.
- 5.
Let \(kT\ln 2=1\).
- 6.
Since the amount of the required free energy (and heat dissipation) is proportional to the length of the best compression of the string, the latter can be seen as a quantification of the erasure transformation’s irreversibility.
- 7.
Since new randomness cannot be gotten rid of later, the equation reads: “Logical reversibility plus randomness equals thermodynamic irreversibility”. If you can go back logically in a random universe, then you certainly cannot thermodynamically.
- 8.
Note that there is no (objective) splitting up, or randomness, if time evolutions are unitary, e.g., come from Schrödinger, heat-propagation, or Maxwell’s equations. What is then the origin of the arrow of time? The quantum-physical version of injectivity is Hugh Everett III’s relative-state interpretation. How do we imagine the bridge from global unitarity to the subjective perception of time asymmetry? When we looked above, with Landauer, at a closed classical system of two parts, then the (possible) complexity deficit in one of them must simply be compensated in a corresponding increase in the other. In Everett’s view, this means that there can be low-entropy branches of the wave function (intuitively, yet too naïvely, called: parallel universes) as long as they are compensated by other, highly complex ones.
- 9.
The counter-factual nature of the reasoning claiming “non-classicality” of quantum theory, that was the main motivation in [38], has already been pointed out by Specker [32]: “In einem gewissen Sinne gehören aber auch die scholastischen Spekulationen über die Infuturabilien hieher, das heisst die Frage, ob sich die göttliche Allwissenheit auch auf Ereignisse erstrecke, die eingetreten wären, falls etwas geschehen wäre, was nicht geschehen ist”. — “In some sense, this is also related to the scholastic speculations on the infuturabili, i.e., the question whether divine omniscience even extends to what would have happened if something had happened that did not happen”.
- 10.
What does it mean that a classical bit exists? Note first that classicality of information implies that it can be measured without disturbance and that the outcome of a “measurement” is always the same; this makes it clear that it is an idealized notion requiring the classical bit to be represented in a redundant way over an infinite number of degrees of freedom, as a thermodynamic limit. It makes thus sense to say that a classical bit U exists, i.e., has taken a definite value.
- 11.
The introduced asymptotic notions are independent of this choice.
- 12.
This is inspired by [9] (see also [10]), where (joint) Kolmogorov complexity — or, in practice, any efficient compression method — is used to define a distance measure on sets of bit strings (such as literary texts of genetic information of living beings). The resulting structure in that case is a distance measure, and ultimately a clustering as a binary tree.
- 13.
Here, h is the binary entropy \(h(x)=-p\log _2 p-(1-p)\log _2(1-p)\). Usually, p is a probability, but h is invoked here merely as an approximation for binomial coefficients.
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Acknowledgments
This text is based on a presentation at the 14th Annual Conference on Theory and Applications of Models of Computation (TAMC 2017) at the Universität Bern. I am grateful to Gerhard Jäger and all the organizers for kindly inviting me to give a talk.
I thank Mateus Araújo, Veronika Baumann, Ämin Baumeler, Charles Bédard, Claus Beisbart, Gilles Brassard, Harvey Brown, Caslav Brukner, Harry Buhrman, Matthias Christandl, Sandro Coretti, Fabio Costa, Bora Dakic, Frédéric Dupuis, Paul Erker, Adrien Feix, Jürg Fröhlich, Manuel Gil, Nicolas Gisin, Esther Hänggi, Arne Hansen, Marcus Huber, Lorenzo Maccone, Alberto Montina, Samuel Ranellucci, Paul Raymond-Robichaud, Louis Salvail, L. Benno Salwey, Martin Schüle, Andreas Winter, and Magdalena Zych for inspiring discussions, and the Pläfä-Einstein as well as the Reitschule for their inspiring atmosphere.
This research is supported by the Swiss National Science Foundation (SNF), the National Centre of Competence in Research “Quantum Science and Technology” (QSIT), the COST action on Fundamental Problems in Quantum Physics, and by the Hasler Foundation.
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Wolf, S. (2017). An All-or-Nothing Flavor to the Church-Turing Hypothesis. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_4
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