Variable-Length Intrinsic Randomness Allowing Positive Value of the Average Variational Distance

This paper considers the problem of variable-length intrinsic randomness. We propose the average variational distance as the performance criterion from the viewpoint of a dual relationship with the problem formulation of variable-length resolvability. Previous study has derived the general formula of the -variable-length resolvability. We derive the general formula of the -variable-length intrinsic randomness. Namely, we characterize the supremum of the mean length under the constraint the value of the average variational distance is smaller than or equal to some constant. Our result clarifies a dual relationship between the general formula of -variable-length resolvability and that of -variable-length intrinsic randomness. We also derive a lower bound of the quantity characterizing our general formula.