Switching boxes connections in operational theories and its consequence on causality

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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.