Limitations and improvements of chain rules for conditional entropies in quantum cryptography security proofs

Security proofs in quantum cryptography rely on conditional entropies.

Chain rules address this problem by relating the conditional entropy of a structured, but non-i.i.d. process to a sum of entropy contributions from each round.

Surprisingly, we show that a natural tightening of the chain rule of Dupuis et al. that would answer this question affirmatively cannot hold, highlighting a limitation of the current DI security proof approach.

Nonetheless, we show that an intermediate improvement is possible by proving a new chain rule in this setting.

In addition, we provide a self-contained framework that unifies existing chain rules and compares their applications, framing our results in a broader context.