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 (4)

• Constructing N-qubit entanglement monotones from anti-linear operators
• An elementary formula for entanglement entropies of fermionic systems
• On the geometry of a class of N-qubit entanglement monotones
• Polynomial identities for hypermatrices

Cited by in corpus (6)

• On polynomial invariants of several qubits
• Permutation-invariant monotones for multipartite entanglement characterization
• Geometry of Two-Qubit State and Intertwining Quaternionic Conformal Mapping Under Local Unitary Transformations
• Invariant and polynomial identities for higher rank matrices
• A Complete Set of Local Invariants for a Family of Multipartite Mixed States
• Bipartite entanglement and control in multiqubit systems