On the geometry of four qubit invariants

Details On the geometry of four qubit invariants On the geometry of four qubit invariants

2006-05-17

arxiv.org/abs/quant-ph/0605151v1

J. Phys. A: Math. Gen. 39 (2006) 9533-9545

Physics

Quantum Physics

19 pages

Scientific paper

10.1088/0305-4470/39/30/009

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 ${\bf CP}^3$. 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 ${\bf CP}^3$ furnished with a fixed quadric.

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