Law of Excluded Quantum Gambling Strategies

Abstract

We introduce and analyze a quantum analogue of the Law of Excluded Gambling Strategies of Classical Decision Theory by the definition of different kind of quantum casinos. The necessity of keeping into account entaglement (by the way we give a staightforward generalization of Schmidt's entanglement measure) forces us to adopt the general algebraic language of Quantum Probability Theory whose essential points are reviewed. The Mathematica code of two packages simulating, respectively, classical and quantum gambling is included. The deep link existing between the censorship of winning quantum gambling strategies and the central notion of Quantum Algorithmic Information Theory, namely quantum algorithmic randomness (by the way we introduce and discard the naive noncommutative generalization of the original Kolmogorov definition), is analyzed