Quantum Minimal One Way Information: Relative Hardness and Quantum Advantage of Combinatorial Tasks

Two-party one-way quantum communication has been extensively studied in the recent literature. We target the size of minimal information that is necessary for a feasible party to finish a given combinatorial task, such as distinction of instances, using one-way communication from another party. This type of complexity measure has been studied under various names: advice complexity, Kolmogorov complexity, distinguishing complexity, and instance complexity. We present a general framework focusing on underlying combinatorial takes to study these complexity measures using quantum information processing. We introduce the key notions of relative hardness and quantum advantage, which provide the foundations for task-based quantum minimal one-way information complexity theory.