  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.
