Algorithmic Probability and Wave Function collapse

Assuming that the evolution of the Quantum State of the Universe may be simulated by the Schrödinger equation, Kolmogorov’s theory of Algorithmic Probability provides us with an elegant mathematical description of what a particular physicist observes during a Quantum Measurement.

Aidan Rocke https://github.com/AidanRocke
12-26-2022

Introduction:

Assuming that the evolution of the Quantum State of the Universe may be simulated by the Schrödinger equation, Kolmogorov’s theory of Algorithmic Probability provides us with an elegant mathematical description of what a particular physicist observes during a Quantum Measurement. Interestingly, this description of non-computable measurements is in qualitative agreement with the Von Neumann-Wigner theory of Quantum Measurement.

Breaking the Von Neumann chain:

  1. Through the paradox of the Von Neumann chain, the Von Neumann-Wigner interpretation of Quantum Mechanics posits that a conscious observer must lie beyond Quantum Computations:

If an observer is a purely physical object, a more comprehensive wave function may now be expressed which encompasses both the state of the Quantum system being observed and the state of the observer. The various possible measurements are now in a superposition of states, representing different observations. However, this leads to a problem: you would now need another measuring device to collapse this larger wave function but this would develop into a superposition state. Another device would be needed to collapse this state ad infinitum. This problem-the Von Neumann chain-is an infinite regression of measuring devices whose stopping point is presumed to be the conscious mind.-Aeowyn Kendall

  1. Von Neumann’s theory of Quantum measurement may thus be summarised as follows:

    1. The Quantum state of a system generally evolves smoothly as dictated by the Schrödinger wave equation.

    2. Otherwise, the Quantum State of this system collapses suddenly and sharply due to a conscious observer.

  2. If we consider that there is a Quantum State associated with the Universe and combine this with the Kantian view that the mind interprets the world, then what a person observes may be defined by the Algorithmic Probability:

\[\begin{equation} P(x|\hat{x})= 2^{-K_U(x\hat{x})} \tag{1} \end{equation}\]

where \(\hat{x}\) denotes the Qualia or conscious state of a person and \(x\) denotes the observations of this person. As Kolmogorov Complexity is not computable, what a particular physicist observes during a Quantum experiment may not be determined by a computable function such as the Schrödinger Wave equation. However, \(P\) may be approximated by a computable probability distribution \(\sum_x f(x|\hat{x}) =1\) since:

\[\begin{equation} \mathbb{E}[K_U(x|\hat{x})] = H(x|\hat{x}) + \mathcal{O}(1) \tag{2} \end{equation}\]

This allows a Quantum Physicist to predict the average outcomes of a Quantum measurement from a frequentist perspective although they can’t predict a particular outcome with certainty.

Consequences:

This has a number of important consequences:

  1. Computability defines the epistemological constraint on a Quantum Physicist that can only determine the average frequencies of experimental outcomes.
  2. The process of conscious observation must lie outside the realm of computable functions including the Schrödinger wave equation.
  3. A hypothetical Quantum Computer would need an Oracle in the loop.

References:

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  8. von Neumann, John (1932). Mathematische Grundlagen der Quantenmechanik (Mathematical Foundations of Quantum Mechanics) Princeton University Press., . ISBN 978-0-691-02893-4.

  9. Shannon, Claude E. (1948). A Mathematical Theory of Communication Bell System Technical Journal 27: 379-423. doi:10.1002/j.1538-7305.1948.tb01338.x.

  10. Markus Mueller. Law without law: from observer states to physics via algorithmic information theory. Arxiv. 2020.

  11. Gerard’t Hooft. The Cellular Automaton Interpretation of Quantum Mechanics. Arxiv. 2015.

  12. Wigner, Eugene; Henry Margenau (1967). “Remarks on the Mind Body Question, in Symmetries and Reflections, Scientific Essays”. American Journal of Physics. 35 (12): 1169–1170.

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Citation

For attribution, please cite this work as

Rocke (2022, Dec. 26). Kepler Lounge: Algorithmic Probability and Wave Function collapse. Retrieved from keplerlounge.com

BibTeX citation

@misc{rocke2022algorithmic,
  author = {Rocke, Aidan},
  title = {Kepler Lounge: Algorithmic Probability and Wave Function collapse},
  url = {keplerlounge.com},
  year = {2022}
}