On dimension bounds for quantum systems

Expressions of several capacity regions in quantum information theory involve an optimization over auxiliary quantum registers. Evaluating such expressions requires bounds on the dimension of the Hilbert space of these auxiliary registers, for which no non-trivial technique is known; we lack a quantum analog of the Caratheodory theorem. We argue in this paper that developing a quantum analog of the Caratheodory theorem requires a better understanding of “quantum convexification.” We then proceed by proving a few results about quantum convexification. To prove one of these results, we develop a new non-Caratheodory-type tool which might be useful for bounding the dimension of quantum registers as well as the cardinality of auxiliary classical random variables.