On relations between probabilities under quantum and classical measurements (Q1768339)

This paper concerns two features that are usually supposed to distinguish quantum from classical probabilities. First is the rule for addition of probabilities. It is argued that where events are mutually exclusive, in the sense of not being able to occur simultaneously in a single trial, then the expression for classical and for quantum sums are analogous. Quantum differences arise where events are not ``probabilistic alternatives in the quantum world'' (p. 693). The second topic, discussed in more detail, is the apparent quantum violation of Bayes rule. Section 3.1 presents first a ``quantum stochastic'' analysis of conditionals where measurements are associated with families of operators that provide not only measurement outcomes, represented by the mutually orthogonal projection operators, but all the possible ``posterior quantum states'' (p. 694) that might feature in conditionals representing a sequence of measurements. An expression is derived for quantum conditionals which has von Neumann's projective rule as a special case. Next in section 3.2 conditionals are discussed in a ``classical frequency'' approach, as transitions from one ``context'' or ``preparation procedure'' to another. Here measurements are regarded as ``filters'' over ``a statistical ensemble prepared by the old context'' (p. 698f). A classical frequency analysis follows. Heisenberg and Bohr are cited in assuming the existence of unmeasurable ``hidden'' combined properties and of measurement perturbations. A ``general transformation relation between probabilities'' (p. 702) is derived which again has von Neumann's projective rule as a special case. The paper is characterised by close attention to the quantum formalism and is therefore necessarily rather technical. There is little debate of wider issues but there are references to other works on related topics, including further work by the same authors.