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Discrete algebra and geometry applied to the Pauli group and mutually unbiased bases in quantum information theory

Abstract : For d not a power of a prime, the maximal number of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space is still unknown. In this thesis, we begin by an original building of MUBs by means of Gauss sums, in relation with a family of irreducible representations of the Lie algebra su(2).Then, we sytematically study the possibility of building such bases by means of Pauli operators. 1) The study of the projective line on Zdm shows that, in order to obtain maximal sets of MUBs, tensorial products of these operators are in order. 2) Lagrangian submodules of Zd2n, of which we give a complete classification, account for maximally commuting sets of Pauli operators. This classification enables to know which of these sets are likely to yield unbiased bases. They correspond to Lagrangian half-modules that can be interpreted as the isotropic points of the projective line (P(Mat(n, Zd)²),ω). Hence, we establish an isomorphism between the unbiased bases thus obtained and distant Lagrangian half-modules, which precises by the way the correspondance between Gauss sums and MUBs. 3) Corollaries on the Clifford group and the finite phase space are then developed.Finally, we present some tools inspired by the previous study. We deal with the cross-ratio on the Bloch sphere and projective geometry in higher dimension, Pauli operators with continuous exponents and we compare von Neumann entropy with a determinantal measure of entanglement
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Submitted on : Thursday, July 28, 2011 - 12:58:12 PM
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  • HAL Id : tel-00612229, version 1



Olivier Albouy. Discrete algebra and geometry applied to the Pauli group and mutually unbiased bases in quantum information theory. Other [cond-mat.other]. Université Claude Bernard - Lyon I, 2009. English. ⟨NNT : 2009LYO10077⟩. ⟨tel-00612229⟩



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