The Probabilistic World

Giorgos Eftaxias, Mirjam Weilenmann, Roger Colbeck

Generalized probabilistic theories (GPTs) provide a framework in which a range of possible theories can be examined, including classical theory, quantum theory, and those beyond. In general, enlarging the state space of a GPT leads to fewer possible measurements because the additional states give stronger constraints on the set of effects, the constituents of measurements. This can have implications for information processing. In box world, for example, a GPT in which any no-signaling distribution can be realized, there is no analog of a measurement in the Bell basis and hence the analog of entanglement swapping is impossible. A comprehensive study of measurements on multiple systems in box world has been lacking. Here we consider such measurements in detail, distinguishing those that can be performed by interacting with individual systems sequentially (termed wirings), and the more interesting set of those that cannot. We compute all the possible box-world effects for cases with small numbers of inputs, outputs, and parties, identifying those that are wirings. The large state space of box world leads to a small effect space and hence the effects of box world are widely applicable in GPTs. We also show some possible uses of nonwirings for information processing by studying state discrimination, nonlocality distillation, and the box-world analog of nonlocality without entanglement. Finally, we connect our results to the study of logically consistent classical processes and to the composition of contextuality scenarios. By enhancing understanding of measurements in box world, our results could be useful in studies of possible underlying principles on which quantum theory can be based.

Lazar Mayants

Alexander Ehmann

Ding Jia

A central open problem of quantum physics is to reconcile theory with experience. In this work I present a framework for studying distinct modes of experience in a superposed world. A modes of experience is characterized by how the world, experiences, and options relate to each other by the perceptions, decisions, and actions, as well as by probabilistic rules encoding probabilistic or deterministic correlations among first person experiences. In a toy model, the life expectancies of beings in different candidate modes of experience are compared. It is found that the quantum mode without macroscopic superposition outlives that with macroscopic superposition and that with real amplitudes. These highlight the prospect to explain a mode of experience by its evolutionary advantages.

David J. Bartholomew

Quantum theory provides the most fundamental account of the physical world and it is at that level, one might suppose, that the roots of God's action would be found. The fact that the theory is probabilistic leads to the hope that the uncertainties present will provide enough room for God to manoeuvre. The conclusion of this chapter is that this hope is too optimistic for very much the same reasons as those used in the simpler cases treated in chapter 8. It is argued that the transition from the superimposed states of the theory to the single observed states is most naturally explained by expressing the problem in terms of conditional probabilities. A STATISTICAL APPROACH How God acts in the world, I repeat, has become one of the most discussed questions in the science and religion field – and beyond. As yet, there is no satisfactory answer to that question. This conclusion has been illustrated in the last chapter, on the small scale and on a restricted front. Now we view the problem on a larger canvas to see how the conclusions reached there survive when they are translated to global, or even, cosmic dimensions. If we find that the conclusions of the last chapter carry over, there is little more to be said. However, the transition from the simple and idealised world of the last chapter is not automatic, because it is not clear whether the quantum world is probabilistic in the way that I have been supposing hitherto.