Graph states and the variety of principal minors

Abstract

AbstractIn 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.