Graph states and the variety of principal minors

In Quantum Information theory, graph states are quantum states defined by graphs. In this work we exhibit a correspondence between orbits of graph states and orbits in the variety of binary symmetric principal minors, under the action of SL(2, F-2)(xn) (sic) G(n). First we study the orbits of maximal abelian subgroups of the n-fold Pauli group under the action of C-n(loc) (sic) G(n), where C-n(loc) is the n-fold local Clifford group, and we show that this action corresponds to the natural action of SL(2, F-2)(xn) (sic) G(n) on the variety Z(n) subset of P(F-2(n2)) of principal minors of binary symmetric n x n matrices: the crucial step is in translating the action of SL(2, F-2)(xn) into an action of the local symplectic group Sp(2n)(loc)(F-2). We conclude by showing how the former action restricts onto stabilizer groups, stabilizer states and graph states.