Abstract
The difficulty of explaining non-local correlations in a fixed causal structure sheds new light on the old debate on whether space and time are to be seen as fundamental. Refraining from assuming space-time as given a priori has a number of consequences. First, the usual definitions of randomness depend on a causal structure and turn meaningless. So motivated, we propose an intrinsic, physically motivated measure for the randomness of a string of bits: its length minus its normalized work value, a quantity we closely relate to its Kolmogorov complexity (the length of the shortest program making a universal Turing machine output this string). We test this alternative concept of randomness for the example of non-local correlations, and we end up with a reasoning that leads to similar conclusions as in, but is conceptually more direct than, the probabilistic view since only the outcomes of measurements that can actually all be carried out together are put into relation to each other. In the same context-free spirit, we connect the logical reversibility of an evolution to the second law of thermodynamics and the arrow of time. Refining this, we end up with a speculation on the emergence of a space-time structure on bit strings in terms of data-compressibility relations. Finally, we show that logical consistency, by which we replace the abandoned causality, it strictly weaker a constraint than the latter in the multi-party case.
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Notes
- 1.
This change of perspective reflects the debate, three centuries ago, between Newton and Leibniz on the nature of space and time, in particular on as how fundamental this causal structure is to be considered.
- 2.
In this context and as a reply to [25], we feel that the notion of a choice between different possible futures by an act of free will put forward there is not only hard to formalize but also not much more innocent than Everettian relative states [21]—after all, the latter are real (within their respective branches of the wave function). We have become familiar with the ease of handling probabilities and cease to realize how delicate they are ontologically.
- 3.
It has been argued that quantum theory violates the causal law due to random outcomes of measurements. Hermann [27] argued that the law of causality does not require the past to determine the future, but vice versa. This is in accordance with our view of logical reversibility: There can be information growth, but there can be no information loss.
- 4.
The introduced asymptotic notions are independent of this choice.
- 5.
This is inspired by Cilibrasi and Vitányi [16], 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.
- 6.
The Church-Turing thesis, first formulated by Kleene [28], states that any physically possible process can be simulated by a universal Turing machine.
- 7.
Note that this is the natural way of defining logical reversibility in our setting with a fixed input and output but no sets nor bijective maps between them.
- 8.
A diagonal argument, called Berry paradox, shows that the notion of “description complexity” cannot be defined generally for all strings.
- 9.
Here, h is the binary entropy h(x) = −plogp − (1 − p)log(1 − p). Usually, p is a probability, but h is invoked here merely as an approximation for binomial coefficients.
- 10.
In this section, conditional complexities are understood as follows: In K(x | y), for instance, the condition y is assumed to be the full (infinite) string, whereas the asymptotic process runs over x [n]. The reason is that very insignificant bits of y (intuitively: the present) can be in relation to bits of x (the past) of much higher significance. The past does not disappear, but it fades.
- 11.
Transitivity arises from the assumption of a fixed causal structure within a party, where the input is causally prior to the output.
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Acknowledgements
This text is based on a presentation at the “Workshop on Time in Physics,” organized by Sandra Ranković, Daniela Frauchiger, and Renato Renner at ETH Zurich in Summer 2015.
The authors thank Mateus Araújo, Veronika Baumann, Charles Bédard, Gilles Brassard, Harvey Brown, Caslav Brukner, Harry Buhrman, Matthias Christandl, Sandro Coretti, Fabio Costa, Bora Dakic, Frédéric Dupuis, Paul Erker, Adrien Feix, Jürg Fröhlich, Nicolas Gisin, Esther Hänggi, Arne Hansen, Marcus Huber, Lorenzo Maccone, Alberto Montina, Samuel Ranellucci, Paul Raymond-Robichaud, Louis Salvail, L. Benno Salwey, Andreas Winter, and Magdalena Zych for inspiring discussions, and the Einstein Kaffee as well as the Reitschule Bern for their inspiring atmosphere.—Grazie mille!
The authors thank Claude Crépeau for his kind invitation to present this work, among others, at the 2016 Bellairs Workshop, McGill Research Centre, Barbados.
Our work was 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 the Hasler Foundation.
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Baumeler, Ä., Wolf, S. (2017). Causality–Complexity–Consistency: Can Space-Time Be Based on Logic and Computation?. In: Renner, R., Stupar, S. (eds) Time in Physics. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-68655-4_6
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