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arXiv:1502.01048

On classical finite probability theory as a quantum probability calculus

David Ellerman

Published Feb 3, 2015• 2 citations
arXivHTMLPDF

Abstract

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There have been four previous attempts to develop a quantum-like model with the base field of $\mathbb{C}$ replaced by $\mathbb{Z}_{2}$, but they are all forced into a merely "modal" interpretation by requiring the brackets to take values in $\mathbb{Z}_{2}$ (1 = possible, 0 = impossible). But the usual QM brackets give the "overlap" between states ψ and ϕ, so for subsets S,T of U, the natural definition is $=|S{\cap}T|$. This allows QM/sets to be developed with a full probability calculus that turns out to be the perfectly classical Laplace-Boole finite probability theory. The point is not to clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features (e.g., two-slit experiment) in a classical setting.

DOI
10.48550/arxiv.1502.01048

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This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There have been four previous attempts to develop a quantum-like model with the base field of C\mathbb{C}C replaced by Z2\mathbb{Z}_{2}Z2​, but they are all forced into a merely "modal" interpretation by requiring the brackets to take values in Z2\mathbb{Z}_{2}Z2​ (1 = possible, 0 = impossible). But the usual QM brackets give the "overlap" between states ψ and ϕ, so for subsets S,T of U, the natural definition is =∣S∩T∣=|S{\cap}T|=∣S∩T∣. This allows QM/sets to be developed with a full probability calculus that turns out to be the perfectly classical Laplace-Boole finite probability theory. The point is not to clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features (e.g., two-slit experiment) in a classical setting.
In the tradition of toy models of quantum mechanics in vector spaces over finite fields (e.g., Schumacher and Westmoreland's "modal quantum theory"), one finite field stands out, 2, since vectors over 2 have an interpretation as natural mathematical objects, i.e., sets. This engages a sets-to-vector-spaces bridge that is part of the mathematical folklore to translate both ways between set concepts and vector space concepts. Using that bridge, the mathematical framework of (finite-dimensional) quantum mechanics can be transported down to sets resulting in quantum mechanics over sets or QM/sets. This approach leads to a different treatment of Dirac's brackets than in "modal quantum theory" (MQT), and that gives a full probability calculus (unlike MQT that only has zero-one modalities of impossible and possible). That, in turn, leads to a rather fulsome theory of QM over sets that includes "logical" models of the double-slit experiment, Bell's Theorem, quantum information theory, quantum computing, and much else. Indeed, QM/sets is proposed as the "logic" of QM in the old-fashioned sense of "logic" as giving the simplified essentials of a theory. QM/sets is also a key part of a broader research program to provide an interpretation of QM based on the notion of "objective indefiniteness," a program that grew out the recent development of the logic of partitions mathematically dual to the usual Boolean logic of subsets.
This paper develops a simplified pedagogical or toy model of (finite-dimensional non-relativistic) quantum mechanics (QM). The model uses vector spaces over Z2={0,1}\mathbb{Z}_{2}=\{0,1\}Z2​={0,1}, where the 0,10,10,1-vectors can also be interpreted as sets, so the model is "quantum mechanics over sets" or QM/Sets. At the level of sets, the key notion is that of a partition or, equivalently, an equivalence relation. Partitions are the logical-level notions to model distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. Those pairs of concepts are the key to understanding the non-classical 'weirdness' of QM. The key non-classical notion in QM is the notion of superposition, i.e., the notion of a state that is indefinite between two or more definite- or eigen-states. As Richard Feynman emphasized, all the weirdness of QM is illustrated in the double-slit experiment so the QM/Sets version is used to make the key points. The simplified model provides an explanation and intuitive picture to answer the key question: "How can the particle get to the detection wall without passing through one slit or the other?"
In the paper the basic concepts of extended probability theory are introduced. The basic idea: the concept of an event as a subset of Ωis replaced with the concept of an event as a partition. The partition is any set of disjoint non-empty subsets of Ω(i.e. partition=subset+its decomposition). Interpretation: elements inside certain part are indistinguishable, while elements from different parts are distinguishable. There are incompatible events, e.g {{e1},{e2}} and {e1,e2}. This is logical incompatibility analogical to the impossibility to have and simultaneously not to have the which-way information in the given experiment. The context is the maximal set of mutually compatible events. Each experiment has associated its context. In each context the extended probability is reduced to classical probability. Then the quadratic representation of events, partitions and probability measures is developed. At the end the central concept of quadratic probability spaces (which extend Kolmogorov probability spaces) is defined and studied. In the next paper it will be shown that quantum mechanics can be represented as the theory of Markov processes in the extended probability theory (Einstein's vision of QM).
Introduction Quantum probability is a generalization of both classical probability theory and quantum mechanics that allows to describe the probabilistic aspects of quantum mechanics. This generalization is formulated in two steps. First, the theory is reformulated in terms of algebras of functions on probability spaces. Therefore, the notion of a probability space (Ω, ℱ , P ) is replaced by the pair ( L ∞ (Ω), E (.) = ∫ Ω . d P ) consisting of the commutative von Neumann algebra of bounded random variables and the expectation functional. Then, the commutativity condition is dropped. In this way we arrive at the notion of a (von Neumann) algebraic probability space ( N ,Φ) consisting of a von Neumann algebra N and a normal (faithful tracial) state Φ. As we have seen this includes classical probability spaces in the form ( L ∞ (Ω), E ), it also includes quantum mechanical systems modeled by a Hilbert space H and a pure state ψ ∊ H (or a mixed state ρ ∊ S ( H )), if we take N = B ( H ) and Ф the state defined by Ф( X ) = 〈 ψ, X ψ 〉 (or Ф ( X ) = tr(ρ X )) for X ∊ B ( H ). Note that in this course we shall relax the conditions on N and Φ and work with involutive algebras and positive normalized functionals, that is, *-algebraic probability spaces. A striking feature of quantum probability (also called noncommutative probability) is the existence of several notions of independence. This is the starting point of this course, which intends to give an introduction to the theory of quantum stochastic processes with independent increments. However, before we come to these processes, we will give a general introduction to quantum probability. In Section 1.2, we recall the basic definitions of quantum probability and discuss some fundamental differences between classical probability and quantum probability. In Section 1.3 we address the question ‘Why do we need Quantum Probability?’ We discuss the EPR experiment and the Kochen–Specker Theorem, which show that we cannot model quantum physics with classical probability spaces because the values of observable quantities do not exist unless we specify which measurement we will carry out and which quantities we will determine. In this sense quantum physics requires a more radical description of chance.
(1) Background: Over a dozen constructions have been proposed to untangle the foundations of quantum mechanics (QM). So many irreconcilable schemes raise the suspect that methodological requisites should be discussed before beginning a theoretical construction in QM. In particular, we identify the ensuing issues of methods: formal definitions of particles and waves; incompatible probability theories and the single case problem. (2) Methods of the present inquiry: We tackle the second and third issue with a novel and comprehensive probability theory. Next it applies the theoretical results to quantum physics. (3) Results of the present inquiry: The new probability theory proves how some effects - the impossibility of measuring, the collapse of outcomes, non-locality and counterfactual definiteness - are not exclusive to quantum mechanics but can be observed also in classical context. (4) Conclusions: A probability-based interpretation of quantum mechanics is derived; innovative answers are put forward about quantum dualism, the measurement problem, collapse and the relations between classical and quantum physics.