In Quantum Information theory, graph states are quantum states defined by graphs. In this work we exhibit a correspondence between graph states and the variety of binary symmetric principal minors, in particular their corresponding orbits under the action of SL(2, F 2 ) ×n ⋊ S n . We start by approaching the topic more widely, that is by studying the orbits of stabilizer-state subgroups of the n-fold Pauli group under an action of C loc n ⋊ S n , where C loc n is the n-fold local Clifford group: we derive this action from the natural action of SL(2, F 2 ) ×n ⋊ S n on the variety Z n ⊂ P(F 2 n 2 ) of principal minors of binary symmetric n × n matrices. The crucial step in this correspondence is in translating the latter action into an action of the local symplectic group Sp loc 2n (F 2 ) on the Lagrangian Grassmannian LG F2 (n, 2n). We conclude by studying how the action on the stabilizer-state groups induces onto the stabilizer states, and finally restricting to the case of graph states.

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$$(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n$$ ( 2 , F 2 ) × n ⋊ S n . First we study the orbits of maximal abelian subgroups of the n-fold Pauli group under the action of $${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$$ C n loc ⋊ S n , where $${\mathcal {C}}_n^{\textrm{loc}}$$ C n loc is the n-fold local Clifford group, and we show that this action corresponds to the natural action of SL$$(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n$$ ( 2 , F 2 ) × n ⋊ S n on the variety $${\mathcal {Z}}_n\subset {\mathbb {P}}({\mathbb {F}}_2^{2^n})$$ Z n ⊂ P ( F 2 2 n ) of principal minors of binary symmetric $$n\times n$$ n × n matrices: the crucial step is in translating the action of SL$$(2,{\mathbb {F}}_2)^{\times n}$$ ( 2 , F 2 ) × n into an action of the local symplectic group Sp$$_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$$ 2 n loc ( F 2 ) . We conclude by showing how the former action restricts onto stabilizer groups, stabilizer states and graph states.