We present a universal construction that relates reversible dynamics on open systems to arbitrary dynamics on closed systems: the restriction affine completion of a monoidal restriction category quotiented by well-pointedness. This categorical completion encompasses both quantum channels, via Stinespring dilation, and classical computing, via Bennetts method. Moreover, in these two cases, we show how our construction can be essentially undone by a further universal construction. This shows how both mixed quantum theory and classical computation rest on entirely reversible foundations.
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