dorsal/arxiv
View SchemaOn the geometry of four qubit invariants
| Authors | Péter Lévay |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0605151 |
| URL | https://arxiv.org/abs/quant-ph/0605151 |
| DOI | 10.1088/0305-4470/39/30/009 |
| Journal | J. Phys. A: Math. Gen. 39 (2006) 9533-9545 |
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 ${\bf CP}^3$. 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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"abstract": "The geometry of four-qubit entanglement is investigated. We replace some of\nthe polynomial invariants for four-qubits introduced recently by new ones of\ndirect geometrical meaning. It is shown that these invariants describe four\npoints, six lines and four planes in complex projective space ${\\bf CP}^3$. For\nthe generic entanglement class of stochastic local operations and classical\ncommunication they take a very simple form related to the elementary symmetric\npolynomials in four complex variables. Moreover, their magnitudes are\nentanglement monotones that fit nicely into the geometric set of $n$-qubit ones\nrelated to Grassmannians of $l$-planes found recently. We also show that in\nterms of these invariants the hyperdeterminant of order 24 in the four-qubit\namplitudes takes a more instructive form than the previously published\nexpressions available in the literature. Finally in order to understand two,\nthree and four-qubit entanglement in geometric terms we propose a unified\nsetting based on ${\\bf CP}^3$ furnished with a fixed quadric.",
"arxiv_id": "quant-ph/0605151",
"authors": [
"P\u00e9ter L\u00e9vay"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1088/0305-4470/39/30/009",
"journal_ref": "J. Phys. A: Math. Gen. 39 (2006) 9533-9545",
"title": "On the geometry of four qubit invariants",
"url": "https://arxiv.org/abs/quant-ph/0605151"
},
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