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arXiv:quant-ph/0304026 · Journal of Physics A Mathematical and General

A complete set of covariants of the four qubit system

Emmanuel Briand, Jean-Gabriel Luque, Jean‐Yves Thibon

Published Sep 9, 2003• 52 citations
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Abstract

We obtain a complete and minimal set of 170 generators for the algebra of SL (2, ) × 4 covariants of a binary quadrilinear form. Interpreted in terms of a four qubit system, this describes in particular the algebraic varieties formed by the orbits of local filtering operations in its projective Hilbert space. Also, this sheds some light on the local unitary invariants, and provides all the possible building blocks for the construction of entanglement measures for such a system.

DOI
10.1088/0305-4470/36/38/309

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We obtain a complete and minimal set of 170 generators for the algebra of SL (2, ) × 4 covariants of a binary quadrilinear form. Interpreted in terms of a four qubit system, this describes in particular the algebraic varieties formed by the orbits of local filtering operations in its projective Hilbert space. Also, this sheds some light on the local unitary invariants, and provides all the possible building blocks for the construction of entanglement measures for such a system.
We obtain a complete and minimal set of 170 generators for the algebra of $SL(2,\C)^{\times 4}$-covariants of a binary quadrilinear form. Interpreted in terms of a four qubit system, this describes in particular the algebraic varieties formed by the orbits of local filtering operations in its projective Hilbert space. Also, this sheds some light on the local unitary invariants, and provides all the possible building blocks for the construction of entanglement measures for such a system.
We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by Stochastic Local Operations with Classical Communication (SLOCC) invariant algebraic varieties. The normal forms of the four-qubit classification of Verstraete et al. are interpreted as dense subsets of components of the dual variety of the set of separable states and an algorithm based on the invariants/covariants of the four-qubit quantum states is proposed to identify a state with a SLOCC equivalent normal form (up to qubits permutation).
It is a recent observation that entanglement classification for qubits is closely related to local SL(2,C)-invariants including the invariance under qubit permutations [Dür, et al., Phys. Rev. A 62, 062314 (2000); Osterloh, A. and Siewert, J., Phys. Rev. A 72, 012337 (2005); O. Chterental and D. Ž. Ðoković, Linear Algebra Research Advances (Nova Science, Hauppauge, N.Y., 2007), Chap. 4, p. 133], which has been termed SL∗ invariance. In order to single out the SL∗ invariants, we analyze the SL(2,C)-invariants of four (five) qubits and decompose them into irreducible modules for the symmetric group S4 (S5) of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of SL∗-invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced by Osterloh and Siewert [Phys. Rev. A 72, 012337 (2005); Int. J. Quant. Inf. 4, 531 (2006)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the Ω-process [Luque, J.-G. and Thibon, J.-Y., J. Phys. A 39, 371 (2005)]. As the degrees of invariants increase, the alternative method proves to be particularly efficient.
An intriguing correspondence between four-qubit systems and simple singularity of type D 4 is established. We first consider the algebraic variety X of separable states within the projective Hilbert space . Then, cutting X with a specific hyperplane H , we prove that the X -hypersurface, defined from the section X ∩ H ⊂ X , has an isolated singularity of type D 4 ; it is also shown that this is the ‘worst-possible’ isolated singularity that one can obtain by this construction. Moreover, it is demonstrated that this correspondence admits a dual version, by proving that the equation of the dual variety of X , which is nothing but the Cayley hyperdeterminant of type 2 × 2 × 2 × 2, can be expressed in terms of the stochastic local operation and classical communication-invariant polynomials as the discriminant of the miniversal deformation of the D 4 -singularity. As a consequence of this correspondence one obtains a finer-grained classification of entanglement classes of four-qubit systems.