On Dimension Bounds for Auxiliary Quantum Systems

Understand

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 Carath\'{e}odory theorem.
• In this paper, we develop a new non-Carath\'{e}odory-type tool for evaluating expressions involving a single quantum auxiliary register and several classical random variables.
• As we show, such expressions appear in problems of entanglement-assisted Gray-Wyner and entanglement-assisted channel simulation, where the question of whether entanglement helps in these settings is related to that of evaluating expressions with a single quantum auxiliary register.