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Algebraic invariants of five qubits

Jean-Gabriel Luque, Jean‐Yves Thibon

Published Jun 7, 2005• 38 citations
OpenAlex

Abstract

The Hilbert series of the algebra of polynomial invariants of pure states of five qubits is obtained, and the simplest invariants are computed.

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Algebraic invariants of five qubits

Jean-Gabriel Luque, Jean‐Yves Thibon

We consider the action of the group on fifth rank tensors on a two-dimensional space, that is, the Hilbert space of a five-qubit system. The Hilbert series of the algebra of polynomial invariants of five qubits pure states is obtained, and the simplest invariants are computed.

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Unitary invariants of qubit systems

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We give an algorithm allowing to construct bases of local unitary invariants of pure k-qubit states from the knowledge of polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are explicited and compared to various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.

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Unitary invariants of qubit systems

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We give an algorithm allowing the construction of bases of local unitary invariants of pure k -qubit states from a knowledge of the polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are made explicit and compared with various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.

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Polynomial invariants of four qubits

Jean-Gabriel Luque, Jean‐Yves Thibon

We describe explicitly the algebra of polynomial functions on the Hilbert space of four-qubit states that are invariant under the group $\mathrm{SL}(2,C{)}^{4}$ of stochastic local quantum operations assisted by classical communication. From this description, we obtain a closed formula for the hyperdeterminant in terms of low degree invariants.

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On polynomial invariants of several qubits

D. Ž. Ðoković, A. Osterloh

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.

arXiv:0804.1661•Journal of Mathematical Physics•Mar 1, 2009•93 citations

Computational aspects of invariants of multipartite quantum systems

Markus Grassl

We consider the problem of finding complete sets of polynomial invariants of multipartite quantum system with respect to local unitary operations. One of the main tools in that respect are both univariate and multivariate Hilbert series which are already hard to compute. We will also discuss various techniques for showing completeness of a set of invariants. Furthermore, new results for three qubit systems will be presented.

•Nov 1, 2009•1 citation
We describe explicitly the algebra of polynomial functions on the Hilbert space of four-qubit states that are invariant under the group SL(2,C)4\mathrm{SL}(2,C{)}^{4}SL(2,C)4 of stochastic local quantum operations assisted by classical communication. From this description, we obtain a closed formula for the hyperdeterminant in terms of low degree invariants.
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.