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

Michel R. P. Planat

Published Nov 30, 2009• 0 citations
OpenAlex

Abstract

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.

Related papers

Balanced Tripartite Entanglement, the Alternating Group A4 and the Lie Algebra sl(3, C) ⊕ u(1)

Michel R. P. Planat, Péter Lévay, Метод Санига

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, 2009•0 citations

Three-qubit entangled embedding of CPT and Dirac groups within E8 Weyl group

Michel R. P. Planat

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.

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Balanced tripartite entanglement, the alternating group A4 and the lie algebra sl(3,ℂ)⊕u(1)

Michel R. P. Planat, Péter Lévay, Метод Санига

arXiv:0912.0172•Reports on Mathematical Physics•Feb 1, 2011•1 citation

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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.

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Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates

Michel R. P. Planat

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$\pm$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.

arXiv:0904.3691•Apr 23, 2009•0 citations

Deterministic transformations of three-qubit entangled pure states

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The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely, the Greenberger-Horne-Zeilinger (GHZ) type and the $W$ type. A state belonging to one of these classes can be stochastically transformed only into a state within the same class by local operations and classical communications. We provide local quantum operations, consisting of the most general two-outcome measurement operators, for the deterministic transformations of three-qubit pure states in which the initial and the target states are in the same class. We explore these transformations, originally having standard GHZ and standard $W$ states, under the local measurement operations carried out by a single party and $p$ ($p=2,3$) parties (successively). We find a notable result that the standard GHZ state cannot be deterministically transformed to a GHZ-type state in which all its bipartite entanglements are nonzero, i.e., a transformation can be achieved with unit probability when the target state has at least one vanishing bipartite concurrence.

arXiv:1905.06631•Physical Review A•Aug 16, 2019•6 citations
Balanced tripartite entanglement, the alternating group A4A_4A4​\\ and the Lie algebra sl(3,C)⊕u(1)sl(3,\mathbb{C})\oplus u(1)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)W(E_8)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)W(E_8)W(E8​) to the four-letter alternating group A4A_4A4​, obtained from a chain of maximal subgroups. Group A4A_4A4​ is realized from two B-type generators and also corresponds to the Lie algebra sl(3,C)⊕u(1)sl(3,\mathbb{C})\oplus u(1)sl(3,C)⊕u(1). Applications are to particle and gravitation physics, and to the genetic code.
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.
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.
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±\pm±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.
The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely, the Greenberger-Horne-Zeilinger (GHZ) type and the WWW type. A state belonging to one of these classes can be stochastically transformed only into a state within the same class by local operations and classical communications. We provide local quantum operations, consisting of the most general two-outcome measurement operators, for the deterministic transformations of three-qubit pure states in which the initial and the target states are in the same class. We explore these transformations, originally having standard GHZ and standard WWW states, under the local measurement operations carried out by a single party and ppp (p=2,3p=2,3p=2,3) parties (successively). We find a notable result that the standard GHZ state cannot be deterministically transformed to a GHZ-type state in which all its bipartite entanglements are nonzero, i.e., a transformation can be achieved with unit probability when the target state has at least one vanishing bipartite concurrence.