Conditional expectation and Bayes' rule for quantum random variables and positive operator valued measures (Q2860747)

The present work represents a step in the development of what the authors call ``quantum probability theory''. This consists in a certain combination of the measure theory of operator valued functions with the theory of positive operator valued measures.NEWLINENEWLINEConcretely, consider a triple \((X,\mathcal{O}(X),\nu)\), where \(X\) is a set, \(\mathcal{O}(X)\) a \(\sigma\)-algebra on \(X\) and \(\nu\) a finite measure valued in positive operators on a finite-dimensional complex Hilbert space \(\mathcal{H}\). Let \(\mu=\mathrm{tr}(\nu)\) be the induced positive measure. By a version of the Radon-Nikodým theorem, there is a measurable function \(h:X\to \mathcal{B}(\mathcal{H})\) with values in positive operators such that \(\nu(A)=\int_A h\, d\mu\) for \(A\in\mathcal{O}(X)\). Take now the square root \(h^{1/2}:X\to \mathcal{B}(\mathcal{H})\) (in the sense of positive operators). Then, for a suitably integrable function \(\psi:X\to \mathcal{B}(\mathcal{H})\), define the integral with respect to \(\nu\) by \(\int \psi\, d\nu := \int h^{1/2} \psi\, h^{1/2}\, d\mu\), see [\textit{D. R. Farenick} and \textit{F. Zhou}, J. Math. Anal. Appl. 327, No. 2, 919--929 (2007; Zbl 1120.26010)].NEWLINENEWLINEThis notion of integral is taken to define a ``quantum expected value'' and developed in the present article in the context of a ``quantum probability theory'' in analogy to ordinary probability theory. In the authors' words, ``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.''