### 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 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, suitable powers of 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 CP^{3} furnished with a fixed quadric.

Original language | English |
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Article number | 009 |

Pages (from-to) | 9533-9545 |

Number of pages | 13 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 39 |

Issue number | 30 |

DOIs | |

Publication status | Published - Jul 28 2006 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

*Journal of Physics A: Mathematical and General*,

*39*(30), 9533-9545. [009]. https://doi.org/10.1088/0305-4470/39/30/009