Characterising transformations between quantum objects, 'completeness' of quantum properties, and transformations without a fixed causal order

Understand

Many fundamental and key objects in quantum mechanics are linear mappings between particular affine/linear spaces.

• This structure includes basic quantum elements such as states, measurements, channels, instruments, non-signalling channels and channels with memory, and also higher-order operations such as superchannels, quantum combs, n-time processes, testers, and process matrices which may not respect a definite causal order.
• Deducing and characterising their structural properties in terms of linear and semidefinite constraints is not only of foundational relevance, but plays an important role in enabling the numerical optimisation over sets of quantum objects and allowing simpler connections between different concepts and objects.
• Here, we provide a general framework to deduce these properties in a direct and easy to use way.