2 research outputs found
On the geometry of four qubit invariants
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 . 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 -qubit ones
related to Grassmannians of -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 furnished with a fixed quadric.Comment: 19 page