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Conditional expectation and Bayes’ rule for quantum random variables and positive operator valued measures

### Abstract

A quantum probability measure ν is a function on a σ-algebra of subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a measure, but where the values of ν are positive operators acting on a complex Hilbert space, and a quantum random variable is a measurable operator valued function. Although quantum probability measures and random variables are used extensively in quantum mechanics, some of the fundamental probabilistic features of these structures remain to be determined. In this paper, we take a step toward a better mathematical understanding of quantum random variables and quantum probability measures by introducing a quantum analogue for the expected value of a quantum random variable ψ relative to a quantum probability measure ν. In so doing we are led to theorems for a change of quantum measure and a change of quantum variables. We also introduce a quantum conditional expectation which results in quantum versions of some standard identities for Radon-Nikodým derivatives. This allows us to formulate and prove a quantum analogue of Bayes’ rule.

© 2012 American Institute of Physics

Received 04 December 2011
Accepted 26 March 2012
Published online 11 April 2012

Acknowledgments:
The work of the authors is supported, in part, by the Natural Sciences and Engineering Research Council of Canada. The second author thanks the Australian National University for its hospitality during his visit from January to May 2011 to the Mathematical Sciences Institute where much of the background material for the present paper was learned.

Article outline:

I. INTRODUCTION
II. QUANTUM CONDITIONAL EXPECTATION
A. Motivating concept: Quantum averaging
B. Measure and integration
C. Properties of quantum expectation
III. CALCULUS
A. Radon-Nikodým theorem
B. Change of quantum measure
C. Change of quantum variables
D. Chain rules
IV. QUANTUM CONDITIONAL EXPECTATION AND BAYES’ RULE
A. Quantum conditional expectation
B. Quantum Bayes’ rule
C. Quantum conditional Jensen's inequality
V. COMPUTING THE QUANTUM CONDITIONAL EXPECTATION: AN EXAMPLE

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2012-04-11

2016-10-01

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