In a recent paper, we have considered the problem of gambling on a quantum experiment and enforce rational behaviour by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In the quantum setting, they yield the Bayesian theory generalised to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we have derived all its four postulates from the generalised Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes’ rule (measurement), marginalisation (partial tracing), independence (tensor product). To say it in a nutshell, we have obtained that quantum mechanics is the Bayesian theory in the complex numbers.
Using this theory we were able to extend Gleason theorem to any dimension by imposing rationality.
- Quantum mechanics: The Bayesian theory generalized to the space of Hermitian matrices. In: Phys. Rev. A, 94 , pp. 042106, 2016.
- A Gleason-Type Theorem for Any Dimension Based on a Gambling Formulation of Quantum Mechanics. In: Foundations of Physics, pp. 1–12, 2017, ISSN: 1572-9516.
- Quantum rational preferences and desirability. In: Proceedings of The 1st International Workshop on “Imperfect Decision Makers: Admitting Real-World Rationality”, NIPS 2016, 2016.