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Non commutative conditional expectations, prediction and a new look at some quantum paradoxes

Henryk Gzyl

Published Feb 15, 2008• 0 citations
OpenAlex

Abstract

In classical probability theory, the best predictor of a future observation of a random variable $X,$ is its expected value $E_P[X]$ when no other information is available, whereas if information consisting in the observation of another random variable is available, then the best predictor is is another random variable $E_P[X|Y].$ It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. we shall see that when the result of an observation is taken into account by means of a non-commutative conditional expectation, exactly as in classical prediction theory, some of the usual paradoxes cease to be so. The moral of this note is that the mystery of the probabilistic interpretation of quantum mechanics lies in the superposition principle

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In classical probability theory, the best predictor of a future observation
of a random variable X,X,X, is its expected value EP[X]E_P[X]EP​[X] when no other
information is available, whereas if information consisting in the observation
of another random variable is available, then the best predictor is is another
random variable EP[X∣Y].E_P[X|Y].EP​[X∣Y]. It is the purpose of this note to explore the
analogue of this in the case of quantum mechanics. we shall see that when the
result of an observation is taken into account by means of a non-commutative
conditional expectation, exactly as in classical prediction theory, some of the
usual paradoxes cease to be so. The moral of this note is that the mystery of
the probabilistic interpretation of quantum mechanics lies in the superposition
principle
In classical probability theory, the best predictor of a future observation of a random variable X,X,X, is its expected value EP[X]E_P[X]EP​[X] when no other information is available When information consisting in the observation of another random variable YYY is available, then the best predictor of XXX is another random variable EP[X∣Y].E_P[X|Y].EP​[X∣Y]. It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. We shall see that exactly as in classical prediction theory, when the result of an observation is taken into account by means of a non-commutative conditional expectation, some of the usual paradoxes cease to be such.
In this note we discuss a few simple classical (as opposed to quantum) prediction problems. The thrust of this work is to examine the predictive role of probability theory and we shall see that some situations are not re-ally paradoxical. In particular we want to separate the role of probability as a predictive tool from one of the basic sources of strangeness in quan-tum mechanics, namely the principle of superposition of states. For that we shall re-examine some of the quantum paradoxes, recast as a problem about making a prediction about the result of a random experiment in classical physics when some information is known. 1
Abstract Given the state of a quantum system, we define the conditional expectation of an observable with respect to a second observable, or commuting family of observables, as the real-valued function of the latter which best approximates the observable, in the sense of minimizing the quantum mechanical expectation of the difference squared. This corresponds to the notion of best predictor of a random variable by another random variable in non-parametric statistical regression. A basic example is the conditional expectation of the quantum mechanical momentum with respect to position, which can be identified with the Bohm momentum. We re-examine the de Broglie–Bohm theory from this perspective. As a second example we consider the simplest quantum system, that of a spin- 1 / 2 paticle in a magnetic field. We study the conditional expectations of the Pauli spin-operators S 1 , S 2 , S 3 with respect to S 3 , and determine their dynamics if the pure-state wave-function solves the relevant Schrödinger equation. Considering S 3 as a beable we can, following Bell, introduce a generalized guidance law in the form of a Markov jump process on the spectrum of S 3 which is compatible with the time evolution of the Born probabilities of the two eigenstates of S 3 . The conditional expectations of the remaining spin operators are then shown to follow a piece-wise deterministic Markov process, defined by an explicit SDE. We finally examine the relationship between the conditional expectations of this paper, and the von Neumann-algebraic conditional expectation onto the sub-algebra generated by the commuting family of conditioning operators when the joint spectrum of the latter is discrete, and derive a general expression for the conditional expectation dynamics in that context.
The aim of this exposé is to make explicit the analogy between the classical notion of non-independent probability distribution and the quantum notion of entangled state. To bring that analogy forth, we consider a classical systems with two dependent random variables and a quantum system with two components. In the classical case, afet observing one of the random variables, the underlying sample space and the probability distribution change. In the quantum case, when and event pertaining to one of the components is observed, the post-measurement state captures, both, the change in the state of the system and implicitly the new probability distribution. The predictions after a measurement in the classical case and in the quantum case, have to be computed with the conditional distribution given the value of the observed variable.
Given a normalized state-vector ψψψ, we define the conditional expectation $\mathbb{E }_{ψ} (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to ψψψ, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.
A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its consequences in the probabilistic description. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables (like in quantum mechanics). Here we propose a method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability. This will be done using the entropic uncertainty relations between random variables, in order to evaluate the presence of non-commutativity in their algebraic description.