Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)

Abstract

We investigate the geometry of the four qubit systems by means of algebraic geometry and invariant theory, which allows us to interpret certain entangled states as algebraic varieties. More precisely we describe the nullcone, i.e., the set of states annihilated by all invariant polynomials, and also the so called third secant variety, which can be interpreted as the generalization of GHZ-states for more than three qubits. All our geometric descriptions go along with algorithms which allow us to identify any given state in the nullcone or in the third secant variety as a point of one of the 47 varieties described in the paper. These 47 varieties correspond to 47 non-equivalent entanglement patterns, which reduce to 15 different classes if we allow permutations of the qubits.

arXiv comment48 pages, 7 tables, 13 figures, references and remarks added (v2)

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Cited by in corpus (17)

• Quantifying entanglement resources
• Entanglement of three-qubit random pure states
• 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
• Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry
• Singularity of type $D_4$ arising from four qubit systems
• Three-qutrit entanglement and simple singularities
• An Algebraic-Geometric Characterization of Tripartite Entanglement
• Geometric approach to entanglement quantification with polynomial measures
• Learning Algebraic Models of Quantum Entanglement
• Classification of multipartite systems featuring only $|W⟩$ and $|GHZ⟩$ genuine entangled states
• Quantum circuits of CNOT gates
• Sequential generation of Polynomial Invariants and N-body non-local correlations
• Local unitary symmetries and entanglement invariants
• Maximally entangled real states and SLOCC invariants: the 3-qutrit case
• Classifying Entanglement by Algebraic Geometry
• Toward Jordan Decompositions of Tensors