Resource Quantification for the No-Programing Theorem

The no-programing theorem prohibits the existence of a universal programmable quantum processor. This statement has several implications in relation to quantum computation but also to other tasks of quantum information processing, making this construction a central notion in this context. Nonetheless, it is well known that, even when the strict model is not implementable, it is possible to conceive of it in an approximate sense. Unfortunately, the minimal resources necessary for this aim are still not completely understood. Here, we investigate quantitative statements of the theorem, improving exponentially previous bounds on the resources required by such a hypothetical machine. The proofs exploit a new connection between quantum channels and embeddings between Banach spaces which allows us to use classical tools from geometric Banach space theory in a clean and simple way.