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A refinement of the argument of local realism versus quantum mechanics by algorithmic randomness

Kohtaro Tadaki

TL;DR

The paper tackles how to operationally interpret probability in quantum mechanics by reframing measurement outcomes through Martin-Löf randomness and the principle of typicality within the many-worlds framework. It refines the classic Bell-CHSH and GHZ analyses by embedding them in an ML-randomness-based probabilistic structure, deriving a QM-consistent equality $\langle RS\rangle+\langle QS\rangle+\langle RT\rangle-\langle QT\rangle=2\sqrt{2}$ and the GHZ perfect correlations, while showing that local realism cannot reproduce these predictions under the same randomness formalism. The work details explicit unitary implementations of Bell and GHZ experimental steps and demonstrates how typical worlds (ML-random) realize QM statistics, strengthening the case for the principle of typicality as a unifying refinement of quantum measurement postulates. Overall, the results offer an operational, randomness-based account that aligns local-realism analyses with QM predictions, with potential implications for interpretations and experimental tests in quantum foundations.

Abstract

The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In our former works [K. Tadaki, arXiv:1804.10174], based on the toolkit of algorithmic randomness, we presented a refinement of the Born rule, called the principle of typicality, for specifying the property of results of measurements in an operational way. In this paper, we make an application of our framework to the argument of local realism versus quantum mechanics for refining it, in order to demonstrate how properly our framework works in practical problems in quantum mechanics.

A refinement of the argument of local realism versus quantum mechanics by algorithmic randomness

TL;DR

The paper tackles how to operationally interpret probability in quantum mechanics by reframing measurement outcomes through Martin-Löf randomness and the principle of typicality within the many-worlds framework. It refines the classic Bell-CHSH and GHZ analyses by embedding them in an ML-randomness-based probabilistic structure, deriving a QM-consistent equality and the GHZ perfect correlations, while showing that local realism cannot reproduce these predictions under the same randomness formalism. The work details explicit unitary implementations of Bell and GHZ experimental steps and demonstrates how typical worlds (ML-random) realize QM statistics, strengthening the case for the principle of typicality as a unifying refinement of quantum measurement postulates. Overall, the results offer an operational, randomness-based account that aligns local-realism analyses with QM predictions, with potential implications for interpretations and experimental tests in quantum foundations.

Abstract

The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In our former works [K. Tadaki, arXiv:1804.10174], based on the toolkit of algorithmic randomness, we presented a refinement of the Born rule, called the principle of typicality, for specifying the property of results of measurements in an operational way. In this paper, we make an application of our framework to the argument of local realism versus quantum mechanics for refining it, in order to demonstrate how properly our framework works in practical problems in quantum mechanics.
Paper Structure (37 sections, 9 theorems, 148 equations)

This paper contains 37 sections, 9 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Contributions of the paper
  3. Organization of the paper
  4. Mathematical preliminaries
  5. Basic notation and definitions
  6. Martin-Löf randomness with respect to an arbitrary probability measure
  7. Martin-Löf $P$-randomness
  8. The properties of Martin-Löf $P$-randomness
  9. The law of large numbers
  10. Non-occurrence of an elementary event with probability zero
  11. Conditional probability
  12. Independence of Martin-Löf $P$-random infinite sequences
  13. Marginal distribution
  14. Postulates of quantum mechanics
  15. The principle of typicality
  16. ...and 22 more sections

Key Result

Theorem 4.1

Let $\Omega$ be an alphabet, and let $P\in\mathbb{P}(\Omega)$. For every $\alpha\in\Omega^\infty$, if $\alpha$ is Martin-Löf $P$-random then for every $a\in\Omega$ it holds that where $N_a(\sigma)$ denotes the number of the occurrences of $a$ in $\sigma$ for every $a\in\Omega$ and every $\sigma\in\Omega^*$. ∎

Theorems & Definitions (19)

  • Definition 2.1: Martin-Löf randomness with respect to a probability measure
  • Definition 3.1: Finite probability space
  • Definition 3.2: Martin-Löf $P$-randomness, Martin-Löf M66
  • Theorem 4.1: The law of large numbers, Tadaki T14
  • Theorem 4.2
  • Corollary 4.3
  • Theorem 4.4: Closure property under conditioning, Tadaki T14
  • Definition 4.5: Independence of Martin-Löf $P$-random infinite sequences, Tadaki T15
  • Theorem 4.6: Tadaki T18arXiv
  • Theorem 4.7
  • ...and 9 more