We characterize and classify quantum correlations in two-fermion systems having $2K$ single-particle states. For pure states we introduce the Slater decomposition and rank (in analogy to Schmidt decomposition and rank); i.e., we decompose the state into a combination of elementary Slater determinants formed by pairs of mutually orthogonal single-particle states. Mixed states can be characterized by their Slater number which is the minimal Slater rank required to generate them. For $K=2\mathrm{}$ we give a necessary and sufficient condition for a state to have a Slater number 1. We introduce a correlation measure for mixed states which can be evaluated analytically for $K=2.\mathrm{}$ For higher $K,$ we provide a method of constructing and optimizing Slater number witnesses, i.e., operators that detect Slater numbers for some states.

We derive a lower bound for the concurrence of mixed bipartite quantum states, valid in arbitrary dimensions. As a corollary, a weaker, purely algebraic estimate is found, which detects mixed entangled states with a positive partial transpose.