Direct trace‑distance converse without purified‑distance smoothing

Establish whether a converse proof for quantum privacy amplification under trace distance can be derived without invoking purified‑distance smoothing; specifically, prove a direct trace‑distance monotonicity argument that holds for all hash functions and yields an upper bound on extractable randomness formulated purely in terms of trace‑distance–based quantities such as the measured smooth min‑entropy H_min^M(X|E).

Background

The paper’s one-shot converse bounds for privacy amplification under trace distance rely on an intermediate step that uses purified distance to obtain a suitable monotonicity under all hash functions. This introduces purified‑distance–smoothed min‑entropy into the analysis of trace‑distance security.

A purely trace‑distance–based converse would avoid this detour and yield bounds that align more directly with the standard cryptographic security criterion. The authors highlight the conceptual importance of removing reliance on purified distance in the converse and pose this as an open question.