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 graph states and the variety of binary symmetric principal minors, in particular their corresponding orbits under the action of $SL(2,\mathbb F_2)^{\times n}\rtimes \mathfrak S_n$. We start by approaching the topic more widely, that is by studying the orbits of maximal abelian subgroups of the $n$-fold Pauli group under the action of $\mathcal C_n^{\text{loc}}\rtimes \mathfrak S_n$, where $\mathcal C_n^{\text{loc}}$ is the $n$-fold local Clifford group: we show that this action corresponds to the natural action of $SL(2,\mathbb F_2)^{\times n}\rtimes \mathfrak S_n$ on the variety $\mathcal Z_n\subset \mathbb P(\mathbb F_2^{2^n})$ of principal minors of binary symmetric $n\times n$ matrices. The crucial step in this correspondence is in translating the action of $SL(2,\mathbb F_2)^{\times n}$ into an action of the local symplectic group $Sp_{2n}^{\text{loc}}(\mathbb F_2)$ on the Lagrangian Grassmannian $LG_{\mathbb F_2}(n,2n)$. We conclude by studying how the former action restricts onto stabilizer groups and stabilizer states, and finally what happens in the case of graph states.