On the geometry of four qubit invariants

Abstract

The geometry of four-qubit entanglement is investigated. We replace some of the polynomial invariants for four-qubits introduced recently by new ones of direct geometrical meaning. It is shown that these invariants describe four points, six lines and four planes in complex projective space ${\mathbf{CP}}^3$. For the generic entanglement class of stochastic local operations and classical communication they take a very simple form related to the elementary symmetric polynomials in four complex variables. Moreover, their magnitudes are entanglement monotones that fit nicely into the geometric set of $n$-qubit ones related to Grassmannians of $l$-planes found recently. We also show that in terms of these invariants the hyperdeterminant of order 24 in the four-qubit amplitudes takes a more instructive form than the previously published expressions available in the literature. Finally in order to understand two, three and four-qubit entanglement in geometric terms we propose a unified setting based on ${\mathbf{CP}}^3$ furnished with a fixed quadric.

19 pages

References in corpus (21)

• Entanglement of Formation of an Arbitrary State of Two Qubits
• Three qubits can be entangled in two inequivalent ways
• Distributed Entanglement
• Four qubits can be entangled in nine different ways
• Global entanglement in multiparticle systems
• Exact and Asymptotic Measures of Multipartite Pure State Entanglement
• Geometric Quantum Mechanics
• A potential multipartide entanglement measure
• Normal forms and entanglement measures for multipartite quantum states
• Classification of multipartite entangled states by multidimensional determinants
• Constructing N-qubit entanglement monotones from anti-linear operators
• The polynomial invariants of four qubits
• Geometry of entangled states, Bloch spheres and Hopf fibrations
• Three-qubit pure-state canonical forms
• An elementary formula for entanglement entropies of fermionic systems
• A generalized skew information and uncertainty relation
• The twistor geometry of three-qubit entanglement
• CP^n, or, entanglement illustrated
• On the geometry of a class of N-qubit entanglement monotones
• A bipartite class of entanglement monotones for N-qubit pure states
• Polynomial identities for hypermatrices

Cited by in corpus (42)

• Quantum entanglement
• Absolute Maximal Entanglement and Quantum Secret Sharing
• Quantifying entanglement resources
• SLOCC classification of n qubits invoking the proportional relationships for spectrums and for standard Jordan normal forms
• The black-hole/qubit correspondence: an up-to-date review
• On polynomial invariants of several qubits
• Classification of Entanglement in Symmetric States
• STU Black Holes as Four Qubit Systems
• Quantification of entanglement via uncertainties
• Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)
• Possibility of generalized monogamy relations for multipartite entanglement beyond three qubits
• Qubits from extra dimensions
• Embedding qubits into fermionic Fock space, peculiarities of the four-qubit case
• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states
• Local Unitary Invariants for N-qubit Pure State
• Two-Center Black Holes, Qubits and Elliptic Curves
• Hyperdeterminants as integrable discrete systems
• A brief introduction to multipartite entanglement
• The attractor mechanism as a distillation procedure
• Permutation-invariant monotones for multipartite entanglement characterization
• Singularity of type $D_4$ arising from four qubit systems
• The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group SL(3,C)
• Unitary Invariants and Classification of Four-Qubit States via Negativity Fonts
• Classification of Bipartite and Tripartite Qutrit Entanglement under SLOCC
• Geometry of pure states of N spin-J system
• Geometry of Two-Qubit State and Intertwining Quaternionic Conformal Mapping Under Local Unitary Transformations
• Nilpotent polynomials approach to four-qubit entanglement
• Two and Three-Qubits Geometry, Quaternionic and Octonionic Conformal Maps, and Intertwining Stereographic Projection
• Stochastic local operations and classical communication equations and classification of even $n$ qubits
• Vector Properties of Entanglement in a Three-Qubit System
• Sequential generation of Polynomial Invariants and N-body non-local correlations
• Relation Between Stereographic Projection and Concurrence Measure in Bipartite Pure States
• Invariant and polynomial identities for higher rank matrices
• Quantum entanglement, Calabi-Yau manifolds, and noncommutative algebraic geometry
• Canonical Forms of $2\times 2\times 2$ and $2\times 2\times 2\times 2$ Symmetric Tensors over Prime Fields
• A Complete Set of Local Invariants for a Family of Multipartite Mixed States
• The necessary and sufficient condition for perfect teleportation and superdense coding and all the suitable states for teleportation and superdense coding
• A study of two-qubit density matrices with fermionic purifications
• Bipartite entanglement and control in multiqubit systems
• Entanglement classification via integer partitions
• Canonical forms of Order-$k$ ($k=2,3,4$) Symmetric Tensors of Format $3\times \cdots \times 3$ Over Prime Fields