[Paper Review] Classical Extensions, Classical Representations and Bayesian Updating in Quantum Mechanics

Guido Bacciagaluppi|ArXiv.org|Mar 7, 2004

Quantum Mechanics and Applications1 references3 citations

TL;DR

This paper explores classical extensions and classical representations of quantum mechanics to model quantum measurement as Bayesian updating. It demonstrates that in the canonical classical extension, quantum state collapse under measurement closely mirrors Bayesian updating with a classical effect, while the uniform classical representation shows a less direct but still meaningful analogy, offering a classical framework for understanding quantum Bayesianism.

ABSTRACT

I review the formalism of classical extensions of quantum mechanics introduced by Beltrametti and Bugajski, and compare it to the classical representations discussed e.g. by Busch, Hellwig and Stulpe and recently used by Fuchs in his discussion of quantum mechanics in terms of standard quantum measurements. I treat the problem of finding Bayesian analogues of the state transition associated with measurement in the canonical classical extension as well as in the related 'uniform' classical representation. In the classical extension, the analogy is extremely good.

Motivation & Objective

• To compare Beltrametti and Bugajski's classical extensions with classical representations in quantum mechanics, particularly in the context of operational quantum theory.
• To investigate whether quantum measurement transitions can be interpreted as Bayesian updating within classical probabilistic frameworks.
• To assess the extent to which quantum state collapse under measurement resembles Bayesian inference in classical systems.
• To clarify the role of non-uniqueness and structure in classical representations and extensions in modeling quantum behavior.

Proposed method

• Uses the convex set formalism to represent quantum states as density operators and observables as affine maps to probability measures.
• Applies the classical extension formalism, where quantum systems are embedded into a larger classical phase space with a surjective reduction map.
• Introduces the concept of a 'canonical' classical extension where quantum measurements correspond to classical Bayesian updating with a classical effect function.
• Analyzes the uniform classical representation, showing that while Bayesian updating is less direct, it still involves a classical update followed by a non-linear disturbance.
• Derives explicit expressions for state transitions after measurement, showing that the updated state corresponds to a Bayesian update with a classical effect, followed by a state-dependent transformation.
• Uses the operator $ A_d $ and the effect $ E_d $ to define a transition rule $ \rho \mapsto \rho^d = \frac{1}{\text{tr}(\rho E_d)} A_d \rho A_d^* $, which is interpreted as Bayesian updating with a classical effect.

Experimental results

Research questions

• RQ1 Can quantum measurement processes be modeled as Bayesian updating within a classical probabilistic framework?
• RQ2 How do classical extensions compare to classical representations in their ability to reproduce quantum measurement statistics?
• RQ3 What is the role of the canonical classical extension in establishing a natural analogy between quantum measurement and Bayesian inference?
• RQ4 How does the non-uniqueness of classical representations affect the interpretation of quantum state collapse as Bayesian updating?
• RQ5 To what extent does the disturbance introduced by measurement in the canonical extension generalize classical Bayes' rule?

Key findings

• In the canonical classical extension, the quantum measurement transition $ \rho \mapsto \rho^d $ corresponds exactly to a Bayesian update $ p(\omega) \mapsto \frac{p(\omega) \langle \omega | E_d | \omega \rangle}{\int p(\omega) \langle \omega | E_d | \omega \rangle d\omega} $, followed by a non-linear, state-independent disturbance.
• The classical effect $ \langle \omega | E_d | \omega \rangle $ depends only on the observable $ E_d $, making the update process interpretable as a classical Bayesian update with a classical effect.
• In the uniform classical representation, the update involves a classical effect $ e_d(\omega) $, but the transition is less directly Bayesian due to the non-positivity of $ e_d^i $ in general.
• The full transition in the canonical extension includes a unitary readjustment, which is essential for recovering the correct quantum state after measurement.
• The analogy between quantum measurement and Bayesian updating is strongest in the canonical classical extension, where the update rule is both operationally and formally analogous to Bayes' rule.
• Despite the classical appearance of the framework, the phase space of composite systems does not factorize as $ \Omega_1 \times \Omega_2 $, indicating non-classical features persist in entangled systems.