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

TL;DR

This paper introduces a new approach to variable-length intrinsic randomness using average variational distance, establishing a dual relationship with variable-length resolvability and deriving a general formula for its characterization.

Contribution

It derives the general formula for $oldsymbol{ ext{epsilon}}$-variable-length intrinsic randomness and clarifies its dual relationship with variable-length resolvability.

Findings

Established the general formula for $oldsymbol{ ext{epsilon}}$-variable-length intrinsic randomness.

Characterized the supremum of mean length under average variational distance constraints.

Derived a lower bound for the key quantity in the formula.

Abstract

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 $\epsilon$-variable-length resolvability. We derive the general formula of the $\epsilon$-variable-length intrinsic randomness. Namely, we characterize the supremum of the mean length under the constraint that the value of the average variational distance is smaller than or equal to a constant $\epsilon$. Our result clarifies a dual relationship between the general formula of $\epsilon$-variable-length resolvability and that of $\epsilon$-variable-length intrinsic randomness. We also derive a lower bound of the quantity characterizing our general formula.