Graph States and the Variety of Principal Minors

Abstract

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 $SL(2,{\mathbb{F}}_2{)}^{\times n}⋊{\mathfrak{S}}_n$. 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 ${\mathcal{C}}_n^{\text{loc}}⋊{\mathfrak{S}}_n$, where ${\mathcal{C}}_n^{\text{loc}}$ is the $n$-fold local Clifford group: we show that this action corresponds to the natural action of $SL(2,{\mathbb{F}}_2{)}^{\times n}⋊{\mathfrak{S}}_n$ 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 $S{p}_{2n}^{\text{loc}}({\mathbb{F}}_2)$ on the Lagrangian Grassmannian $L{G}_{{\mathbb{F}}_2}(n,2n)$. 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.

40 pages, 2 tables

References in corpus (9)

• Experimental demonstration of graph-state quantum secret sharing
• Hyperdeterminantal relations among symmetric principal minors
• Optimal preparation of graph states
• Optimal verification of stabilizer states
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Synthesis of Logical Clifford Operators via Symplectic Geometry
• The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy
• Lagrangian Grassmannians and Spinor Varieties in Characteristic Two