The reconstruction of phylogenetic trees from mixed populations has become important in the study of cancer evolution, as sequencing is often performed on bulk tumor tissue containing mixed populations of cells. Recent work has shown how to reconstruct a perfect phylogeny tree from samples that contain mixtures of two-state characters, where each character/locus is either mutated or not. However, most cancers contain more complex mutations, such as copy-number aberrations, that exhibit more than two states. We formulate the Multi-State Perfect Phylogeny Mixture Deconvolution Problem of reconstructing a multi-state perfect phylogeny tree given mixtures of the leaves of the tree. We characterize the solutions of this problem as a restricted class of spanning trees in a graph constructed from the input data, and prove that the problem is NP-complete. We derive an algorithm to enumerate such trees in the important special case of cladisitic characters, where the ordering of the states of each character is given. We apply our algorithm to simulated data and to two cancer datasets. On simulated data, we find that for a small number of samples, the Multi-State Perfect Phylogeny Mixture Deconvolution Problem often has many solutions, but that this ambiguity declines quickly as the number of samples increases. On real data, we recover copy-neutral loss of heterozygosity, single-copy amplification and single-copy deletion events, as well as their interactions with single-nucleotide variants.