On Dimension Bounds for Auxiliary Quantum Systems

Abstract

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 Carathéodory theorem. In this paper, we develop a new non-Carathéodory-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. To evaluate such expressions, we argue that developing a quantum analog of the Carathéodory theorem requires a better understanding of a notion which we call ``quantum conditioning." We then proceed by proving a few results about quantum conditioning, one of which is that quantum conditioning is strictly richer than the usual classical conditioning.

30 pages, title changed, structure significantly improved, results unchanged, to appear in IEEE TIT

References in corpus (6)

• Coding Theorem and Strong Converse for Quantum Channels
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• The Communication Cost of Simulating Bell Correlations
• Remote preparation of quantum states
• The quantum capacity with symmetric side channels
• Quantifying the nonlocality of GHZ quantum correlations by a bounded communication simulation protocol

Cited by in corpus (7)

• On contraction coefficients, partial orders and approximation of capacities for quantum channels
• Channel Simulation and Coded Source Compression
• Hadamard quantum broadcast channels
• Bounds on Information Combining With Quantum Side Information
• The Entropy Power Inequality with quantum conditioning
• Additivity of quantum capacities in simple non-degradable quantum channels
• Squashed quantum non-Markovianity: a measure of genuine quantum non-Markovianity in states