Entanglement Classification with Algebraic Geometry

Abstract

We approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry, its natural language. We show that the class of symmetric separable states has the structure of a Veronese variety and that its $k$-secant varieties are SLOCC invariants. Thus SLOCC classes gather naturally into families. This classification presents useful properties such as a linear growth of the number of families with the number of particles, and nesting, i.e. upward consistency of the classification. We attach physical meaning to this classification through the required interaction length of parent Hamiltonians. We show that the states $W_N$ and GHZ${}_N$ are in the same secant family and that, effectively, the former can be obtained in a limit from the latter. This limit is understood in terms of tangents, leading to a refinement of the previous families. We compute explicitly the classification of symmetric states with $N\le 4$ qubits in terms of both secant families and its refinement using tangents. This paves the way to further use of projective varieties in algebraic geometry to solve open problems in entanglement theory.

References in corpus (12)

• Experimental quantum teleportation
• Environment-Assisted Quantum Walks in Photosynthetic Energy Transfer
• Inductive Entanglement Classification of Four Qubits under SLOCC
• Matrix Product States: Symmetries and Two-Body Hamiltonians
• Inductive classification of multipartite entanglement under SLOCC
• Tripartite entanglement transformations and tensor rank
• Artificial Life in Quantum Technologies
• Geometric descriptions of entangled states by auxiliaries varieties
• Entanglement, fractional magnetization and long-range interactions
• SLOCC Invariants for Multipartite Mixed States
• Entanglement classification with matrix product states
• Entanglement of three-qubit geometry

Cited by in corpus (9)

• Characterizing multipartite entanglement classes via higher-dimensional embeddings
• The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing
• Quantum Entanglement involved in Grover's and Shor's algorithms: the four-qubit case
• Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry
• Multipartite quantum correlations: symplectic and algebraic geometry approach
• Non-gaussian Entanglement Swapping between Three-Mode Spontaneous Parametric Down Conversion and Three Qubits
• Relations among $k$-ME concurrence, negativity, polynomial invariants, and tangle
• Stellar Representation of Grassmannians
• Classifying Entanglement by Algebraic Geometry