The Asymmetric Divergence

KL divergence $D_{KL}(P|Q)$ is often used to compare two probability distributions, but it is not a true distance metric. Which core metric axiom does it fail to satisfy in general?

It has no failing axiom; KL divergence is a valid metric

Non-negativity: the value can be negative for some distributions

Symmetry: $D_{KL}(P|Q)$ does not generally equal $D_{KL}(Q|P)$

Identity: $D_{KL}(P|P)$ is not zero