Given a fixed causal structure between systems, Bell inequality violations provide a way of certifying that some of those systems must have necessarily been non-classical. Going beyond the notion of a fixed background space-time or fixed causal structures, the notion of indefinite causal orders has been widely studied specially within the process matrix framework . Here, causal inequalities have been proposed as a method for certifying the non-classicality of causal orders, but their physical meaning is relatively less understood. A better understanding of this can be gained by first characterising processes that can be described within a fixed space-time and analysing the assumptions needed to rule out trivial violations of causal inequalities that can be achieved through fixed causal orders. For example, one such assumption is that the input to each local lab precedes its output (“local order”) which is implicit in the process matrix framework. However, modelling this assumption using measurements of local clocks to order these events could decohere the process being considered.
To this effect, I will compare two frameworks for studying causality: the causal boxes framework  and the process matrix framework. The former assumes a fixed background space-time, while the latter does not, yet both can describe the quantum switch (QS) which is claimed to be an example of an indefinite causal order. After reviewing the frameworks and the description of QS in both of them, I will compare these descriptions and show that they are inequivalent. There are crucial differences between the two mathematical representations of QS which provides interesting preliminary insights into some of the questions mentioned above, in particular for modelling the “local order” assumption in a framework independent manner. This comparison also reveals the differences between the optical (fixed space-time) and the gravitational (indefinite space- time) implementations of QS.
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