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arXiv:0804.1661 · Journal of Mathematical Physics

On polynomial invariants of several qubits

D. Ž. Ðoković, A. Osterloh

Published Mar 1, 2009• 93 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.3075830

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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.
1. For mathematical point of view, Quantum Information Theory deals with finite dimensional Hilbert spaces, the state spaces of finite k-bipartite systems. In the case of pure k-bipartite systems, this space has the special form 2. H=V1 ⊗ ... ⊗ Vk, 3. where Vi denotes a finite dimensional state space of the ith particle of the system (most of the time, one assumes that it is a qubit, i.e. dim Vi = 2). The interesting non-classical behavior already occurs for two-qubit systems, for the so-called entangled states: the states which cannot be written in the form v1 ⊗ v2. There is no general agreement on the definition of entanglement for system with more than two parts. Klyachko has proposed to regard as entangled the states which are semi-stable for the action of the SLOCC group (G=SL(V1) × ... × SL(Vk)) in the sense of geometric invariant theory. Geometric invariant theory provides 4. methods for characterizing such states without explicitly computing the invariants. In order to test the significance of this property, we have explicitly computed the invariants in the simplest cases. 5. The non-locality properties of an entangled states does not change under unitary operations acting independently on each of its subsystem. The idea to construct entanglement measure by means of unitary invariant have been explored for example by Grassl et al. (1998), Schlienz and Mahler (1996). We have proved in 2006, that such invariants can be deduced from SLOCC-covariants. The knowledge of all the covariants and hence, the classification of the SLOCC orbits seems to be relevant in this context. However, except for simplest systems there are far too many orbits and the complete classification is out of reach. 6. 7. The aim of the series of four lectures given by Jean-Gabriel Luque and Jean-Yves Thibon is to explain a subproblem: how to compute the Hilbert series of the algebraic invariants for the pure k-qubit systems. 8. 9. In the two first lectures, Jean-Gabriel Luque will give a brief account on the symmetric functions and the characters of the symmetric groups. The first two talks will be meant for an audience of beginner in the area of symmetric functions. In the 10. two remaining lectures, Jean-Yves Thibon will explore the links between symmetric functions and Kronecker product (Lecture 3) in order to explain how to compute the Hilbert series (Lecture 4).
The number of entanglement classes in stochastic local operations and classical communication (SLOCC) classifications increases with the number of qubits and is already infinite for four qubits. Criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore at the focus of intense theoretical research. We develop a general criterion for the discrimination of pure $N\text{\ensuremath{-}}$partite entangled states in terms of polynomial $\mathrm{SL}(d,\mathrm{C}){}^{\ensuremath{\bigotimes}N}$ invariants. By means of this criterion, existing SLOCC classifications of four-qubit entanglement are reproduced. Based on this we propose a polynomial classification scheme in which entanglement types are identified through ``tangle patterns.'' This scheme provides a practicable way to classify states of arbitrary multipartite systems. Moreover, the use of polynomials induces a corresponding quantification of the different types of entanglement.
In quantum information theory, understanding the complexity of entangled states within the context of SLOCC (stochastic local operations and classical communications) involving d qubits (or qudits) is essential for advancing our knowledge of quantum systems. This complexity is often analyzed by classifying the states via local symmetry groups. The resulting classes can be distinguished using invariant polynomials, which serve as a measure of entanglement. This paper introduces a novel method for obtaining invariant polynomials of the smallest degrees, which significantly enhances the efficiency of characterizing SLOCC classes of entangled quantum states. Our method not only simplifies the process of identifying these classes but also provides a robust tool for analyzing the entanglement properties of complex quantum systems. As a practical application, we demonstrate the derivation of minimal degree invariants in specific cases, illustrating the effectiveness of our approach in real-world scenarios. This advancement has the potential to streamline various processes in quantum information theory, making it easier to understand, classify, and utilize entangled states effectively.
It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.
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.