Minimax sample-complexity lower bounds for adaptive and collective protocols

Prove a minimax lower bound for estimating the conditional phase face against adaptive, entangled, collective, sparse-recovery, or joint-context protocols, thereby determining whether the stated 4^m variance scaling and separate-context acquisition costs are information-theoretically optimal.

Background

The paper derives a local Cramér–Rao bound with 4^m scaling for a single m-control face under a specified independent, uniform Ramsey acquisition design. It also gives an n·2^n−1+x context factor for an exhaustive audit when separate budgets are assigned to target–context faces.

These results are explicitly protocol-specific rather than universal lower bounds. The unresolved problem is to determine optimal worst-case sample complexity when more powerful strategies—including adaptive, entangled, collective, sparse-recovery, or joint-context protocols—are allowed.