Multi-system measurements in generalized probabilistic theories and their role in information processing

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Abstract
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 boxworld, for example, a GPT in which any no-signalling distribution can be realised, there is no analogue of a measurement in the Bell basis and hence the analogue of entanglement swapping is impossible. A comprehensive study of measurements on multiple systems in boxworld 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 boxworld effects for cases with small numbers of inputs, outputs and parties, identifying those that are wirings. The large state space of boxworld leads to a small effect space and hence the effects of boxworld are widely applicable in GPTs. We also show some possible uses of non-wirings for information processing by studying state discrimination, nonlocality distillation and the boxworld analogue 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 boxworld, our results could be useful in studies of possible underlying principles on which quantum theory can be based.

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