Error Exponents in Distributed Hypothesis Testing of Correlations

We study a distributed hypothesis testing problem where two parties observe i.i.d. samples from two ρ-correlated standard normal random variables X and Y. The party that observes the X-samples can communicate R bits per sample to the second party, that observes the Y-samples, in order to test between two correlation values. We investigate the best possible type-II error subject to a fixed type-I error, and derive an upper (impossibility) bound on the associated type-II error exponent. Our techniques include representing the conditional Y-samples as a trajectory of the Ornstein-Uhlenbeck process, and bounding the associated KL divergence using the subadditivity of the Wasserstein distance and the Gaussian Talagrand inequality.