Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-Löf random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.
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Accepted version, forthcoming in The Proceedings of The 2026 Meeting of the Philosophy of Science Association
Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-Löf random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.
In order to make the right empirical predictions Bohmian mechanics requires a special statistical boundary condition -- the distribution postulate -- but it is unclear how best to understand this condition. We show how one might use the theory of algorithmic randomness to formulate the distribution postulate as an objective constraining law. The framework requires us to say something about admissible quantum-mechanical states and measurements. In return, algorithmic Bohmian mechanics (aBM) guarantees the standard Born statistics for a collection of canonical quantum experiments in the limit, not just with high probability. The algorithmic distribution postulate provides a sharp typicality condition, clarifies the status of quantum probabilities in the deterministic theory, and provides a concrete example of how notions provided by the theory of algorithmic randomness can aid in specifying the content of a physical law.
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 this paper, based on the toolkit of algorithmic randomness, we present a refinement of the Born rule, as an alternative rule to it, for specifying the property of the results of quantum measurements in an operational way. Algorithmic randomness is a field of mathematics which enables us to consider the randomness of an individual infinite sequence. We then present an operational refinement of the Born rule for mixed states, as an alternative rule to it, based on algorithmic randomness. In particular, we give a precise definition for the notion of mixed state. We then show that all of the refined rules of the Born rule for both pure states and mixed states can be derived from a single postulate, called the principle of typicality, in a unified manner. We do this from the point of view of the many-worlds interpretation of quantum mechanics. Finally, we make an application of our framework to the BB84 quantum key distribution protocol in order to demonstrate how properly our framework works in practical problems in quantum mechanics, based on the principle of typicality.
This article presents a survey of published and unpublished material of the intersection of Algorithmic Information Theory and Quantum Mechanics. It is, to the author's knowledge, the first of its type. Three different notions of the algorithmic content of quantum states are reviewed. Notions of algorithmic quantum typicality and mutual information are introduced. The relationship between algorithmic information and quantum measurements is explored. One of the surprising results is that an overwhelming majority of quantum (pure and mixed) states, when undertaking decoherence, will result in a classical probability with no algorithmic information. Thus most quantum states decohere into white noise. A quantum analouge of Martin Löf random sequence is reviewed. Algorithmic Information Theory presents new complications for the Many Worlds Theory, as it conflicts with the Independence Postulate. When algorithmically complicated processes are ruled out, measurements are required to produce distributions over quantum states that have cloneable information.
Abstract Nies and Scholz formalized the notion of an infinite qubitstring and referred to it as a ‘state’. They defined ‘quantum Martin-Löf randomness’ for states. We give a notion of measurement of a state in a computable basis and introduce ‘quantum measurement randomness’, a randomness notion for states. A state is quantum measurement random if measuring it in any computable basis yields a Martin-Löf random bitstring with probability one. Our main result is that quantum Martin-Löf randomness strictly implies quantum measurement randomness. This uses the construction of a quantum measurement random state which is not quantum Martin-Löf random. We prove two general results on which this construction relies: The first concerns Martin-Löf randomness relative to computable measures and extends a result of V. Vovk. The second is a combinatorial result about Kronecker products.
Abstract This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a (Wittgensteinian) philosophical discussion of randomness in general, I argue that deterministic interpretations of quantum mechanics (like Bohmian mechanics or ’t Hooft’s Cellular Automaton interpretation) are strictly speaking incompatible with the Born rule. I also stress the role of outliers, i.e. measurement outcomes that are not 1-random. Although these occur with low (or even zero) probability, their very existence implies that the no-signaling principle used in proofs of randomness of outcomes of quantum-mechanical measurements (and of the safety of quantum cryptography) should be reinterpreted statistically, like the second law of thermodynamics. In three appendices I discuss the Born rule and its status in both single and repeated experiments, review the notion of 1-randomness (or algorithmic randomness) that in various guises was investigated by Kolmogorov and others and treat Bell’s (Physics 1:195–200, 1964) Theorem and the Free Will Theorem with their implications for randomness.
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 Bell's inequality versus quantum mechanics for refining it, in order to demonstrate how properly our framework works in practical problems in quantum mechanics.