Invariant information and complementarity in high-dimensional states

Using a generalization of the invariant information introduced by Brukner and Zeilinger [Phys. Rev. Lett. $\mathbf{\text{83}}$, 3354 (1999)] to high-dimensional systems, we introduce a complementarity relation between the local and nonlocal information for $d\times d$ systems under the isolated environment, where $d$ is prime or the power of prime. We also analyze the dynamics of the local information in the decoherence process.