Fixed-marginal quantum privacy amplification under purified distance

Establish a tight second-order expansion for privacy amplification with quantum side information under the fixed-marginal purified-distance criterion, equivalently the fixed-marginal order-/2 Renyi-divergence criterion.

Background

The paper discusses prior work by Tomamichel and Hayashi on privacy amplification with quantum side information after optimizing the reference side-information state. That result yields a tight second-order expansion for the optimized criterion, but the corresponding fixed-marginal problem remains unresolved. The present paper explains why this problem is difficult even in the commuting case: the fixed-marginal criterion can exhibit a Gaussian-mixture second-order profile rather than the single Gaussian profile arising after marginal optimization.

The unresolved problem concerns determining the exact second-order behavior when the reference side-information state is fixed rather than optimized, particularly for the purified-distance criterion or its equivalent order-1/2 Renyi-divergence formulation.