A Bayesian account of quantum histories

Abstract

We investigate whether quantum history theories can be consistent with Bayesian reasoning and whether such an analysis helps clarify the interpretation of such theories. First, we summarise and extend recent work categorising two different approaches to formalising multi-time measurements in quantum theory. The standard approach consists of describing an ordered series of measurements in terms of history propositions with non-additive 'probabilities'. The non-standard approach consists of defining multi-time measurements to consist of sets of exclusive and exhaustive history propositions and recovering the single-time exclusivity of results when discussing single-time history propositions. We analyse whether such history propositions can be consistent with Bayes' rule. We show that certain class of histories are given a natural Bayesian interpretation, namely the linearly positive histories originally introduced by Goldstein and Page. Thus we argue that this gives a certain amount of interpretational clarity to the non-standard approach. We also attempt a justification of our analysis using Cox's axioms of probability theory.

Quantum mechanics with extended probabilities

“…That difference should not obscure the fact that (1.2) was first used a definition of probability by Goldstein and Page who also discussed of some of its properties. Similar formulae have also been discussed in [6,12,13].…”

“…While individual bettors may want restrict the notion of record by criteria beyond (3.16), there is no reason to incorporate such restrictions in a fundamental formulation of quantum theory. Indeed, there is every reason not to 13 . All the notions (1)-( 5) are human specific and none of them admits of a precise formulation.…”

Bayesian analog of Gleason’s theorem

“…Thus the linearly positive histories and the consistent histories programmes appear naturally within this peg framework. If we wish to discuss real Bayesian probabilities we could follow the linearly positive histories programme and take the real parts of our pegs and ensure a linearly positive condition [22]. Similarly, if we wish to discuss relative frequencies we could follow the consistent histories programme and take the real parts of our complex pegs and define a consistency condition stronger than linear positivity [23].…”

Bayesian Probabilities and the Histories Algebra

“…Such an argument adds weight to Youssef's derivation of quantum-like features-similarly for Caticha's recent pedagogical derivation of a quantum theory (Caticha, 1998). Similarly we could ask the same question for history theories-and Marlow (2006) has shown that there is a subset of histories where the standard Bayesian rules apply-but we already have a complex object that might obey Cox's axioms; namely (8).…”

“…Recent work (Marlow, 2006) has shown that there is a certain amount of consistency between a real notion of Bayesian probability and the LP formalism. These LP probabilities seem to obey Bayesian reasoning whereas the standard notion of probability using the decoherence functional does not.…”