Geometric descriptions of entangled states by auxiliaries varieties

Abstract

The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting we describe well-known classifications of multipartite entanglement such as $2\times 2\times (n+1)$, for $n\ge 1$, quantum systems and a new example with the $2\times 3\times 3$ quantum system. Our description completes the approach of Miyake and makes stronger connections with recent work of algebraic geometers. Moreover for the quantum systems detailed in this paper we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.

arXiv comment32 pages, 15 Tables, 5 Figures. References and remarks added

References in corpus (12)

• Three qubits can be entangled in two inequivalent ways
• Four qubits can be entangled in nine different ways
• Geometric Quantum Mechanics
• Classification of multipartite entangled states by multidimensional determinants
• The polynomial invariants of four qubits
• Inductive Entanglement Classification of Four Qubits under SLOCC
• Four-qubit entanglement from string theory
• Multipartite entanglement in 2 x 2 x n quantum systems
• The moduli space of three qutrit states
• Coherent states, entanglement, and geometric invariant theory
• Classification of multipartite entanglement containing infinitely many kinds of states
• Entanglement of three-qubit geometry

Cited by in corpus (19)

• Quantifying entanglement resources
• Geometry of spin coherent states
• Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)
• Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon
• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• Quantum Entanglement involved in Grover's and Shor's algorithms: the four-qubit case
• Entanglement Classification with Algebraic Geometry
• Singularity of type $D_4$ arising from four qubit systems
• An Algebraic-Geometric Characterization of Tripartite Entanglement
• A link between Quantum Entanglement, Secant varieties and Sphericity
• Dimension of Tensor Network varieties
• Generalized Weyl-Heisenberg algebra, qudit systems and entanglement measure of symmetric states via spin coherent states
• Learning Algebraic Models of Quantum Entanglement
• Classification of multipartite systems featuring only $|W⟩$ and $|GHZ⟩$ genuine entangled states
• Hyperdeterminants from the $E_8$ Discriminant
• Stellar Representation of Grassmannians
• Classifying Entanglement by Algebraic Geometry
• Maximally entangled real states and SLOCC invariants: the 3-qutrit case
• Toward Jordan Decompositions of Tensors