Switching boxes connections in operational theories and its consequence on causality

Abstract

How can we describe a device that takes two unknown operational boxes and conditionally on some input variable connects them in different orders? In order to answer this question, I will introduce maps from transformations to transformations within operational probabilistic theories with purification, and show their characterisation in terms of operational circuits. I will then proceed exploring the hierarchy of maps on maps. A particular family of maps in the hierarchy are the ones whose output is in the set of transformations. These maps can be fully characterised by their correspondence with channels with memory, and it is exactly the family of transformations that can be implemented through operational circuits. I will then show the problems that arise in defining the remainder of the hierarchy, and the reason why we cannot avoid considering its elements. The main consequence of admitting the generalised transformations as possible within the operational theory is that we cannot describe them in terms of simple causal connection of transformations in a circuit with a fixed causal structure. In quantum theory, we can understand such higher order transformations in terms of superpositions of circuits with different causal structures. The problem whether computations exploiting higher-order transformations can be efficiently simulated by a conventional circuital computer is posed.