Non-commutative Bayesian expectation and its connection to quantum theory

S. Sýkora

Yasmín Navarrete, Sergio M. Davis

An understanding of quantum theory in terms of new, underlying descriptions capable of explaining the existence of non-classical correlations, non-commutativity of measurements and other unique and counter-intuitive phenomena remains still a challenge at the foundations of our description of physical phenomena . Among some proposals, the idea that quantum states are essentially states of knowledge in a Bayesian framework is an intriguing possibility due to its explanatory power. In this work, the formalism of quantum theory is derived from the application of Bayesian probability theory to “fragile” systems, that is, systems that are perturbed by the measurement. Complex Hilbert spaces , non-commuting operators and the trace rule for expectations all arise naturally from the use of linear algebra to solve integral equations involving classical probabilities over hidden variables. The non-fragile limit of the theory, where all measurements are commutative and the theory becomes analogous to classical statistical theory is discussed as well.

Yasmín Navarrete, Sergio M. Davis

The formalism of Quantum Mechanics is derived from the application of Bayesian probability theory to fragile systems, i.e. systems that are perturbed by the act of measurement. Complex Hilbert spaces, non-commuting operators and the trace rule for expectations all arise naturally from the use of linear algebra to solve integral equations involving classical probabilities over hidden variables. We comment on the case of non-local hidden variables, where violations of Bell's theorem can be produced, as well as the non-fragile limit of the theory, where all measurements are commutative and the theory becomes analogous to classical statistical mechanics.

Werner Stulpe

Miklós Rédei

Based on recalling two characteristic features of Bayesian statistical inference in commutative probability theory, a stability property of the inference is pointed out, and it is argued that that stability of the Bayesian statistical inference is an essential property which must be preserved under generalization of Bayesian inference to the non‐commutative case. Mathematical no‐go theorems are recalled then which show that, in general, the stability can not be preserved in non‐commutative context. Two possible interpretations of the impossibility of generalization of Bayesian statistical inference to the non‐commutative case are offered, none of which seems to be completely satisfying.

Christopher A. Fuchs, Rüdiger Schack