We study the sample complexity of conditional independence testing. In this problem, given i.i.d. samples from a discrete distribution $P_{ABC}$, the goal is to distinguish whether $A$ and $C$ are conditionally independent with respect to $B$, i.e., $P_{ABC}=P_{A|B}P_BP_{C|B}$, or whether $A$ and $C$ are conditionally dependent, $\Delta(P_{ABC},P_{A|B}P_BP_{C|B})\geq \varepsilon$ for some fixed threshold $\varepsilon$ and distance measure $\Delta$. We are interested in the cases where $\Delta$ is either the $\ell_1$ distance or the KL-divergence. The study for the case of $\ell_1$ distance was initiated by (Canonne et al., STOC 2018), and the KL-divergence was recently studied by (Seyfried et al., COLT 2025). Both works design algorithms whose sample complexities scale sublinearly in the dimensions of the subsystems, and showed tight lower bounds in some parameter regimes. While Canonne et al. derived partial lower bounds for the remaining regimes as well, the problem of fully resolving the sample complexity in all parameters remained open. In this work, we settle these open questions and prove optimal sample complexity lower bounds for both of these problems, thereby completely settling the sample complexities up to polylogarithmic factors.

We investigate the sample complexity of mutual information and conditional mutual information testing. For conditional mutual information testing, given access to independent samples of a triple of random variables $(A, B, C)$ with unknown distribution, we want to distinguish between two cases: (i) $A$ and $C$ are conditionally independent, i.e., $I(A:C|B) = 0$, and (ii) $A$ and $C$ are conditionally dependent, i.e., $I(A:C|B) \geq \varepsilon$ for some threshold $\varepsilon$. We establish an upper bound on the number of samples required to distinguish between the two cases with high confidence, as a function of $\varepsilon$ and the three alphabet sizes. We conjecture that our bound is tight and show that this is indeed the case in several parameter regimes. For the special case of mutual information testing (when $B$ is trivial), we establish the necessary and sufficient number of samples required up to polylogarithmic terms. Our technical contributions include a novel method to efficiently simulate weakly correlated samples from the conditionally independent distribution $P_{A|B} P_{C|B} P_B$ given access to samples from an unknown distribution $P_{ABC}$, and a new estimator for equivalence testing that can handle such correlated samples, which might be of independent interest.