Cohomology in algebra, geometry, physicsand statistics

Classification of four qubit and rebit states

We consider the problem of classifying the orbits of SL(2, C) ^4 on the space
C^2 ⊗ C^2 ⊗ C^2 ⊗ C^2. In quantum information theory this is known as the
classification of four qubit states under SLOCC operations. We approach
the problem by constructing the representation via a symmetric pair of max-
imal rank. This makes it possible to apply the theory of θ-representations
developed by Vinberg in the 70’s. The orbits are devided into three types:
nilpotent, semisimple and mixed. The orbits of each type are classified sep-
arately. We also obtain the stabilizers of representatives of the orbits. The
talk will end with some comments on the same problem over R, known as
the classification of four rebit states. This is joint work with Heiko Dietrich,
Alessio Marrani and Marcos Origlia.

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