On the Geometry and Invariants of Qubits, Quartits and Octits (original) (raw)
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Peres/Mermin arguments about no-hidden variables in quantummechanics are used for displaying a pair (R,S) of entangling Clifford quantum gates, acting on two qubits. From them, a natural unitary realization 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 group dipoles C± n. The union of the three-qubit real Clifford group C+3 and the Toffoli gate ensures a orthogonal realization 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, W and chain families. PACS numbers: 03.67.Pp, 03.67.Pp, 02.20.-a, 03.65.Ud 1.
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Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates
General Mathematics Notes
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 realization of Coxeter/Weyl groups W (D 5 ) and W (F 4 ) emerges, which is also reflected into the splitting of the n-qubit Clifford group C n into group dipoles C ± n . The union of the three-qubit real Clifford group C + 3 and the Toffoli gate ensures a orthogonal realization of the Weyl/Coxeter group W (E 8 ), and of its relatives. Other concepts involved are complex reflection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.
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We have given a novel formulation of the exact solutions for the lambda, vee and cascade three-level systems where the Hamiltonian of each configuration is expressed in the SU(3) basis. The solutions are discussed from the perspective of the Bloch equation and the atom-field entanglement scenario. For the semiclassical systems, the Bloch space structure of each configuration is studied by solving the corresponding Bloch equation and it is shown that at resonance, the eight-dimensional Bloch sphere is broken up into two distinct subspaces due to the existence of a pair of quadratic constants. Because of the different structure of the Hamiltonian in the SU(3) basis, the non-linear constants are found to be distinct for different configurations. We propose a possible representation of the qutrit wave function and show its equivalence with the three-level system. Taking the bichromatic cavity modes to be in the coherent state, the amplitudes of all three quantized systems are calculated by developing an Euler angle based dressed state scheme. Finally following the Phoenix-Knight formalism, the interrelation between the atom-field entanglement and population inversion for all configurations is studied and the existence of collapses and revivals of two different types is pointed out for the equidistant cascade system in particular.