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- Switching boxes connections in operational theories and its consequence on causality

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10020071

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.

©2012 Perimeter Institute for Theoretical Physics