Concurrence in arbitrary dimensions

We argue that a complete characterization of quantum correlations in bipartite systems of many dimensions may require a quantity which, even for pure states, does not reduce to a single number. Subsequently, we introduce multidimensional generalizations of concurrence and find evidence that they may provide useful tools for the analysis of quantum correlations in mixed bipartite states. We also introduce biconcurrence that leads to a necessary and sufficient condition for separability.