Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R;S) of entangling Clifiord quantum gates, acting on two qubits. From them, a unitary realization of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also re∞ected into the splitting of the nqubit Clifiord group Cn into subgroup dipoles C § . Generators of the three-qubit real Clifiord group C + 3 with the Tofioli gate ensures a orthogonal realization of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex re∞ection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.
Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R;S) of entangling Clifiord quantum gates, acting on two qubits. From them, a unitary realization of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also re∞ected into the splitting of the nqubit Clifiord group Cn into subgroup dipoles C § . Generators of the three-qubit real Clifiord group C + 3 with the Tofioli gate ensures a orthogonal realization of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex re∞ection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.
Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R, S) of entangling Clifford quantum gates, acting on two qubits. From them, a natural unitary representation of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also reflected into the splitting of the n-qubit Clifford group Cn into dipoles C±n . The union of the three-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonal representation of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex reflection groups, BN pairs, unitary group designs and entangled states of the GHZ family.
In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group P, and two-qubit Pauli group P2, respectively. It has been found [M. Socolovsky, Int. J. Theor. Phys. 43, 1941 (2004)] that the CPT group of the Dirac equation is isomorphic to P. One introduces a two-qubit entangling orthogonal matrix S basically related to the CPT symmetry. With the aid of the two-qubit swap gate, the S matrix allows the generation of the three-qubit real Clifford group and, with the aid of the Toffoli gate, the Weyl group W(E8) is generated (M. Planat, Preprint 0904.3691). In this paper, one derives three-qubit entangling groups ˜ P and ˜ P2, isomorphic to the CPT group P and to the Dirac group P2, that are embedded into W(E8). One discovers a new class of pure theequbit quantum states with no-vanishing concurrence and three-tangle that we name CPT states. States of the GHZ and CPT families encode the new representation of the Dirac group and its CPT subgroup.
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.
Dans un premier temps, je pr´esenterais bri`evement comment David Mermin rend compte des aspects paradoxaux de la r´ealit´e quantique, `a partir de la non-compatibilit´e alg´ebrique des r`egles pour les valeurs propres (mesures quantiques) et pour les vecteurs propres (´etats quantiques) d'ensembles d'observables qui ne commutent pas tous. Je montrerai alors comment on peut transf´erer un tel questionnement `a un groupe `a sym´etrie octahedrale et `a ses invariants. Ensuite, j'introduirais au calcul quantique, o`u les portes unitaires sont les ´el´ements d'un groupe ad´equat. On cherche `a contenir les erreurs du calcul, repr´esent´ees par les produits tensoriels des matrices de Pauli, dans un sous-groupe fini de portes quantiques, ´el´ements du groupe unitaire complexe U(2n) `a n qubits, et `a repr´esenter au mieux toute matrice unitaire (universalit´e du calcul), d'o`u le groupe de Clifford et sa relation au groupe symplectique Sp(2n, 2). La d´ecomposition de Sp(2n, 2) en ses sous-groupes orthogonaux (extrasp´eciaux) se refl`ete en une d´ecomposition du groupe de Clifford en deux dipoles, dont les ´el´ements rel`event de plusieurs concepts int´eressants: les groupes de r´eflections complexes, les designs unitaires, ainsi que l'intrication quantique. Je les introduirai en d´etail dans le s´eminaire. Puis je montrerai que le groupe de Coxeter/Weyl fini le plus grand W(E8), d'ordre 696 729 600, peux-ˆetre repr´esent´e par des portes quantiques r´eelles `a trois qubits, dont l'une au moins, la porte de Toffoli, est ´etrang`ere au groupe de Clifford. La structure de W(E8), ainsi repr´esent´ee, est intimement li´ee au ´etats GHZ, et sa d´ecomposition en sous-groupes maximaux est li´ee aux ´etats bipartites W ou aux ´etats `a intrication en chaˆıne. Quelques perspectives pour le calcul quantique et l'unification de la physique seront discut´ees. Bibliographie Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates. Preprint 0904.3691
The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group and two extra axioms are satisfied by the double cosets acting on B. The BN-pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits (R Jozsa and A Miyake, Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are related to the configuration with 27 lines on a smooth cubic surface.
Balanced tripartite entanglement, the alternating group A4\\ and the Lie algebra sl(3,C)⊕u(1)
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(E8) 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(E8) to the four-letter alternating group A4, obtained from a chain of maximal subgroups. Group A4 is realized from two B-type generators and also corresponds to the Lie algebra sl(3,C)⊕u(1). Applications are to particle and gravitation physics, and to the genetic code.