Equality in the probabilistic variational representation of Sibson’s α-mutual information for finite α

Determine whether equality can be attained in the probabilistic variational representation of Sibson’s α-mutual information given in Theorem thm:markovChainRepresentation for any finite α>1. Specifically, prove or disprove that for every joint distribution P_{XY} there exists an auxiliary random variable U satisfying the Markov chain U−X−Y such that exp(((α−1)/α) I_α(X,Y)) equals the supremum over U of E_{P_Y}[||P_{U|Y}||_{L^α(P_U)}] divided by || ||P_{U|X}||_{L^β(P_X)} ||_{L^α(P_U)}, where β=α/(α−1); equivalently, characterize conditions or constructions of P_{U|X} that achieve equality, or show that equality cannot be achieved for some P_{XY}.

Background

The paper introduces a probabilistic variational representation aiming to mirror the operational form used for maximal leakage (α=∞). For finite α>1, Theorem thm:markovChainRepresentation establishes only an inequality: exp(((α−1)/α) I_α(X,Y)) ≤ sup_{U:U−X−Y} E_{P_Y}[||P_{U|Y}||{L^α(P_U)}] / || ||P{U|X}||{L^β(P_X)} ||{L^α(P_U)}. The authors note that for α→∞ (maximal leakage), equality can be achieved using a shattering construction for P_{U|X}, but analogous equality for finite α remains unresolved.

This open problem seeks to clarify whether the bound is tight for finite α, by either constructing an auxiliary variable U and conditional distribution P_{U|X} that achieve equality universally or by identifying counterexamples and conditions under which equality is impossible.