Abstract. We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group W(E8) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar “condensation ” of W(E8) into the four-letter alternating group A4, obtained from a chain of maximal subgroups. Group A4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, C) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned. 1.Abstract. We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group W(E8) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar “condensation ” of W(E8) into the four-letter alternating group A4, obtained from a chain of maximal subgroups. Group A4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, C) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned. 1.

Jan 1, 20090 citations

Study of entanglement for tripartite symmetrical states

Ruben Quesada

Aug 27, 20120 citations

Balanced tripartite entanglement, the alternating group $A_4$ \\ and the Lie algebra $sl(3,\mathbb{C})\oplus u(1)$ Balanced tripartite entanglement, the alternating group $A_4$\\ and the Lie algebra $sl(3,\mathbb{C})\oplus u(1)$

Michel R. P. Planat

I feature three important classes of three-qubit entangled states and their encoding into quantum gates, discrete groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. I introduce another generic class B of three-qubit states, that are equally balanced over two and three parties. I show how to realize the largest cristallographic group $W(E_8)$ in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates}, Preprint 0904.3691 (quant-ph)]. Then, I describe a peculiar condensation of $W(E_8)$ to the four-letter alternating group $A_4$ , obtained from a chain of maximal subgroups. Group $A_4$ is realized from two B-type generators and also corresponds to the Lie algebra $sl(3,\mathbb{C})\oplus u(1)$ . Applications are to particle and gravitation physics, and to the genetic code.I feature three important classes of three-qubit entangled states and their encoding into quantum gates, discrete groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. I introduce another generic class B of three-qubit states, that are equally balanced over two and three parties. I show how to realize the largest cristallographic group $W(E_8)$ in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates}, Preprint 0904.3691 (quant-ph)]. Then, I describe a peculiar condensation of $W(E_8)$ to the four-letter alternating group $A_4$, obtained from a chain of maximal subgroups. Group $A_4$ is realized from two B-type generators and also corresponds to the Lie algebra $sl(3,\mathbb{C})\oplus u(1)$. Applications are to particle and gravitation physics, and to the genetic code.

Nov 30, 20090 citations

Bound Entanglement for Bipartite and Tripartite Quantum Systems

Hui Zhao, Sha Guo

International Journal of Theoretical PhysicsMar 17, 20151 citation

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.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:0804.1661Journal of Mathematical PhysicsMar 1, 200993 citations

Balanced tripartite entanglement, the alternating group

A_4

\\ and the Lie algebra

sl(3,\mathbb{C})\oplus u(1)

W(E_8)

in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates}, Preprint 0904.3691 (quant-ph)]. Then, I describe a peculiar condensation of

W(E_8)

to the four-letter alternating group

A_4

, obtained from a chain of maximal subgroups. Group

A_4

is realized from two B-type generators and also corresponds to the Lie algebra

sl(3,\mathbb{C})\oplus u(1)

. Applications are to particle and gravitation physics, and to the genetic code.