# The Kolmogorov Structure Function and its role in Quantum Measurement

In the following analysis, it is shown that the Kolmogorov Structure Function builds a robust bridge between Occam’s razor and the Universal Wave Function.

Aidan Rocke https://github.com/AidanRocke
06-24-2023

Let’s suppose that the Universal Wave Function determines the evolution of all observables so every observable in an experimental setting, $$\widehat{X}$$, represents a particular realisation of a random variable, $$X$$, that evolves according to the linear Schrödinger equation.

Drawing inspiration from Leonid Levin’s analysis of information-theoretic conservation laws, we may consider the Kolmogorov Structure Function and its possible role in Quantum Measurement:

$\begin{equation} \Phi(X) := \mathbb{E}[K_U(X)] + K_U(\widehat{X}) \tag{1} \end{equation}$

In a Universe governed by the Universal Wave Function, the Von Neumann entropy of the Quantum State of the Universe is invariant to Unitary transformations. In such a Universe that satisfies the Law of Conservation of Information, experiments are repeatable so the entropy term that represents the description length of our stochastic data:

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

is a conserved quantity.

Meanwhile, in a Universe where information is globally conserved, Occam’s razor is applicable and may be formalised via Levin’s Coding theorem:

$\begin{equation} K_U(\widehat{X}) = -\log_2 P(\widehat{X}) - \mathcal{O}(1) \tag{3} \end{equation}$

where the entropy term which describes the description length of the appropriate statistical model, $$-\log_2 P(\widehat{X})$$, is a second conserved quantity.

Finally, our analsis of the Kolmogorov Structure Function may be asymptotically partitioned as a sum of two conserved quantities:

$\begin{equation} \Phi(X) \sim H(X) - \log_2 P(\widehat{X}) \tag{4} \end{equation}$

This analysis is particularly consequential for the Von Neumann-Wigner formulation of Quantum Measurement as well as Markus Müller’s more recent formulation of Quantum Mechanics as a theory of what we observe next. If Qualia plays a role in Quantum Measurements then certain Qualia associated with foresight may lower both entropy terms in a reliable manner, which is verifiable within the setting of iterated Bayesian Games .

If true, one of the more interesting consequences of the Quantum Qualia Hypothesis is that both Classical Machine Learning and Quantum Machine Learning rely upon the same fundamental ontology. In fact, they must be misnomers.

1. Stephen Hawking & James B. Hartle. The Wave Function of the Universe. Physical Review. 1983.

2. Edward Witten. A Mini-Introduction to Information Theory. Arxiv. 2018.

3. Peter Grünwald and Paul Vitanyí. Shannon Information and Kolmogorov Complexity. 2010.

4. L.A. Levin. Laws of information conservation and aspects of the foundation of probability theory. Problems of Information Transmission. 1974.

5. Wigner, Eugene & Henry Margenau. Remarks on the Mind Body Question, in Symmetries and Reflections, Scientific Essays. American Journal of Physics. 1967.

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

7. Nicolas Brunner & Noah Linden. Connection between Bell nonlocality and Bayesian Game theory. Nature Communications. 2013.