What is Quantum in probabilistic explanations of the sure-thing principle violation?

The Prisoner's Dilemma game (PDG) is one of the simple test-beds for the probabilistic nature of the human decision-making process. Behavioral experiments have been conducted on this game for decades and show a violation of the so-called sure-thing principle, a key principle in the rational theory of decision. Quantum probabilistic models can explain this violation as a second-order interference effect, which cannot be accounted for by classical probability theory. Here, we adopt the framework of generalized probabilistic theories and approach this explanation from the viewpoint of quantum information theory to identify the source of the interference. In particular, we reformulate one of the existing quantum probabilistic models using density matrix formalism and consider different amounts of classical and quantum uncertainties for one player's prediction about another player's action in PDG. This enables us to demonstrate that what makes possible the explanation of the violation is the presence of quantum coherence in the player's initial prediction and its conversion to probabilities during the dynamics. Moreover, we discuss the role of other quantum information-theoretical quantities, such as quantum entanglement, in the decision-making process. Finally, we propose a three-choice extension of the PDG to compare the predictive powers of quantum probability theory and a more general probabilistic theory that includes it as a particular case and exhibits third-order interference.

Emergence of correlated proton tunnelling in water ice.

Several experimental and theoretical studies report instances of concerted or correlated multiple proton tunnelling in solid phases of water. Here, we construct a pseudo-spin model for the quantum motion of protons in a hexameric H2O ring and extend it to open system dynamics that takes environmental effects into account in the form of O-H stretch vibrations. We approach the problem of correlations in tunnelling using quantum information theory in a departure from previous studies. Our formalism enables us to quantify the coherent proton mobility around the hexagonal ring by one of the principal measures of coherence, the l 1 norm of coherence. The nature of the pairwise pseudo-spin correlations underlying the overall mobility is further investigated within this formalism. We show that the classical correlations of the individual quantum tunnelling events in long-time limit is sufficient to capture the behaviour of coherent proton mobility observed in low-temperature experiments. We conclude that long-range intra-ring interactions do not appear to be a necessary condition for correlated proton tunnelling in water ice.