Group-theoretical approach to entanglement

We present a universal description of quantum entanglement using group theory and noncommutative characteristic functions. It leads to reformulations of the separability problem, which allows us to generalize the latter, thus connecting the theory of entanglement and harmonic analysis. As an example, we translate and analyze the positivity of partial transpose criterion and a simple criterion for pure states into the group-theoretical language. We also show that when applied to finite groups, our formalism embeds the separability problem in a given dimension into a higher dimensional but highly symmetric one. Finally, our formalism reveals a connection between the very existence of entanglement and group noncommutativity.