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arXiv:1905.03332 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

The Born rule as a statistics of quantum micro-events

Yurii Vladimirovich Brezhnev

Published Dec 1, 2020• 4 citations
arXivHTMLPDF

Abstract

We deduce the Born rule from a purely statistical take on quantum theory within minimalistic math-setup. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics—a linear, not Hilbert’, vector space—and empirical notion of the Statistical Length of a state. Its statistical nature comes from the lab micro-events (detector-clicks) being formalized into the C -coefficients of quantum superpositions. We also comment that not only has the use not been made of quantum axioms (scalar-product, operators, interpretations , etc.), but that the involving thereof would be, in a sense, inconsistent when deriving the rule. In point of fact, the quadratic character of the statistical length, and even not (the ‘physics’ of) Born’s formula, represents a first step in constructing the mathematical structure we name the Hilbert space of quantum states.

DOI
10.1098/rspa.2020.0282

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We deduce the Born rule from a purely statistical take on quantum theory within minimalistic math-setup. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics—a linear, not Hilbert’, vector space—and empirical notion of the Statistical Length of a state. Its statistical nature comes from the lab micro-events (detector-clicks) being formalized into the C -coefficients of quantum superpositions. We also comment that not only has the use not been made of quantum axioms (scalar-product, operators, interpretations , etc.), but that the involving thereof would be, in a sense, inconsistent when deriving the rule. In point of fact, the quadratic character of the statistical length, and even not (the ‘physics’ of) Born’s formula, represents a first step in constructing the mathematical structure we name the Hilbert space of quantum states.
Clarifying the nature of the quantum state |Ψ⟩ is at the root of the problems with insight into counter-intuitive quantum postulates. We provide a direct—and math-axiom free—empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure—quantum superposition—is based on a set-theoretic creation of ensemble formations and invokes the following three principia: (I) quantum statics, (II) doctrine of the number in the physical theory, and (III) mathematization of matching the two observations with each other (quantum covariance). All of the constructs rest upon a formalization of the minimal experimental entity—the registered micro-event, detector click. This is sufficient for producing the ℂ-numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the spatio-temporal concepts. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.
The Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.
The Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.
Finite probability theory is enriched by introducing the mathematical notion (no physics involved) of a superposition event ΣS\Sigma SΣS--in addition to the usual discrete event SSS (subset of the outcome space U=(u1,...,un)U=\left( u_{1},...,u_{n}\right)U=(u1​,...,un​)). Mathematically, the two types of events are distinguished using n×nn\times nn×n density matrices. The density matrix ρ(S)\rho\left( S\right)ρ(S) for a discrete event is diagonal and the density matrix ρ(ΣS)\rho\left( \Sigma S\right)ρ(ΣS) is obtained as an outer product ∣s⟩⟨s∣\left\vert s\right\rangle \left\langle s\right\vert∣s⟩⟨s∣ of a normalized vector ∣s⟩∈Rn\left\vert s\right\rangle \in\mathbb{R}^{n}∣s⟩∈Rn. Probabilities are defined using density matrices as Pr⁡(T∣ρ)=tr⁡[PTρ]\Pr\left( T|\rho\right) =\operatorname*{tr}\left[ P_{T}\rho\right]Pr(T∣ρ)=tr[PT​ρ] where T⊆UT\subseteq UT⊆U and PTP_{T}PT​ is the diagonal projection matrix with diagonal entries χT(ui)\chi_{T}\left( u_{i}\right)χT​(ui​). Then for the singleton {ui}⊆U\left\{ u_{i}\right\} \subseteq U{ui​}⊆U, the probability of the outcome uiu_{i}ui​ conditioned by the _superposition_ event ΣS\Sigma SΣS is Pr⁡({ui}∣ΣS)=⟨ui∣s⟩2\Pr\left( \left\{ u_{i}\right\} |\Sigma S\right) =\left\langle u_{i}|s\right\rangle ^{2}Pr({ui​}∣ΣS)=⟨ui​∣s⟩2, the Born Rule. Thus the Born Rule arises naturally from the mathematics of superposition when superposition events are added to ordinary finite probability theory. No further explanation is required when the mathematics uses Cn\mathbb{C}^{n}Cn instead of Rn\mathbb{R}^{n}Rn except that the square ⟨ui∣s⟩2\left\langle u_{i}|s\right\rangle ^{2}⟨ui​∣s⟩2 is the absolute square ∣⟨ui∣s⟩∣2\left\vert \left\langle u_{i}|s\right\rangle \right\vert ^{2}∣⟨ui​∣s⟩∣2.
The Born rule, a foundational axiom used to deduce probabilities of events from wavefunctions, is indispensable in the everyday practice of quantum physics. It is also key in the quest to reconcile the ostensibly inconsistent laws of the quantum and classical realms, as it confers physical significance to reduced density matrices, the essential tools of decoherence theory. Following Bohr's Copenhagen interpretation, textbooks postulate the Born rule outright. However, recent attempts to derive it from other quantum principles have been successful, holding promise for simplifying and clarifying the quantum foundational bedrock. A major family of derivations is based on envariance, a recently discovered symmetry of entangled quantum states. Here, we identify and experimentally test three premises central to these envariance-based derivations, thus demonstrating, in the microworld, the symmetries from which the Born rule is derived. Further, we demonstrate envariance in a purely local quantum system, showing its independence from relativistic causality.
The Born rule stands as the fundamental probability postulate of quantum mechanics, yet its first-principle derivation remains a longstanding open problem. This paper presents a rigorous, collapse-free, and observer-independent derivation of the Born rule. Starting from four physically natural and minimal axioms—unitary dynamics, uniqueness of classical records, unitary invariance, and consistency with the classical limit and ergodicity—we uniquely deduce the Born rule from first principles. We prove that the Born rule is not an independent postulate, but a necessary mathematical consequence of the core axiomatic structure. Our approach is self-contained, assumption-free, and purely unitary, establishing that the Born rule is canonically determined by the fundamental mathematical and physical principles of quantum mechanics.