Classical Probability and Quantum Outcomes

Abstract

There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.

On Quantum Conditional Probability

“…6 2 The relation between the non-existence of the joint distribution of two observables and their incompatibility is subtle and depends critically on the fact that a joint distribution is defined in terms of a particular state W . See Gudder (1968), Gudder (1979), Malley (2004), Malley and Fine (2005), Malley (2006), Malley and Fletcher (2008), Nelson (1967) and Varadarajan (1962). 3 For a detailed formulation of quantum probability theory see, for example, Beltrametti and Cassinelli (1981), Bub (1974) or Hughes (1989).…”