Tightness of the conditional-dependence converse bound

Determine whether the single-letter upper bound E(R) ≤ \tilde{E}(R) is tight for testing against conditional dependence when the transmitter observes X^n but not the side-information sequence Z^n, beyond the special case in which Z is a deterministic function of X.

Background

The paper studies testing against conditional dependence, in which the receiver observes Y^n and Z^n while the transmitter observes only X^n. Under the null hypothesis, the joint distribution is P_{XZ}P_{Y|Z}; under the alternative, it is Q_{XYZ}.

By introducing a related conditional-testing-against-dependence problem in which Z^n is available to both transmitter and receiver, the paper derives the single-letter exponent \tilde{E}(R). For the original problem, where Z^n is unavailable to the transmitter, this yields only the converse bound E(R) ≤ \tilde{E}(R). The paper proves equality when Z is a function of X, because the transmitter can then reconstruct Z^n, but leaves the tightness question unresolved in the general case.