Graph States and the Variety of Principal Minors
Vincenzo Galgano, Fr\'ed\'eric Holweck

TL;DR
This paper explores the mathematical relationship between quantum graph states and the variety of principal minors of binary symmetric matrices, revealing new symmetries and orbit structures under group actions.
Contribution
It establishes a correspondence between graph states in quantum information and the variety of principal minors, connecting group actions on quantum states to algebraic geometry.
Findings
Identifies the action of local Clifford groups with the action on principal minors.
Translates the action of $SL(2,\mathbb F_2)^{\times n}$ into a symplectic group action.
Analyzes stabilizer groups and states, especially graph states.
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 . We start by approaching the topic more widely, that is by studying the orbits of maximal abelian subgroups of the -fold Pauli group under the action of , where is the -fold local Clifford group: we show that this action corresponds to the natural action of on the variety of principal minors of binary symmetric matrices. The crucial step in this correspondence is in translating the action…
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Taxonomy
TopicsGraph theory and applications · Molecular spectroscopy and chirality · Advanced NMR Techniques and Applications
