Ranee Brylinski

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In this paper we study universality for quantum gates acting on qudits. Qudits are states in a Hilbert space of dimension d where d can be any integer ≥ 2. We determine which 2-qudit gates V have the properties (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit(More)
Let M be a G-covering of a nilpotent orbit in g where G is a complex semisimple Lie group and g = Lie(G). We prove that under Poisson bracket the space R[2] of homogeneous functions on M of degree 2 is the unique maximal semisimple Lie subalgebra of R = R(M) containing g. The action of g ′ ≃ R[2] exponentiates to an action of the corresponding Lie group G ′(More)
In the framework of geometric quantization we explicitly construct, in a uniform fashion, a unitary minimal representation pio of every simply-connected real Lie group Go such that the maximal compact subgroup of Go has finite center and Go admits some minimal representation. We obtain algebraic and analytic results about pio. We give several results on the(More)
We construct a unique G-equivariant graded star product on the algebra S (g)/I of polynomial functions on the minimal nilpotent coadjoint orbit O min of G where G is a complex simple Lie group and g = sl(2, C). This strengthens the result of Arnal, Benamor and Cahen. Our main result is to compute, for G classical, the star product of a momentum function µ x(More)
Let X R be a (generalized) flag manifold of a non-compact real semisimple Lie group G R , where X R and G R have complexifications X and G. We investigate the problem of constructing a graded star product on Pol(T * X R) which corresponds to a G R-equivariant quantization of symbols into smooth differential operators acting on half-densities on X R. We show(More)
Let O be the minimal nilpotent adjoint orbit in a classical complex semisim-ple Lie algebra g. O is a smooth quasi-affine variety stable under the Eu-ler dilation action C * on g. The algebra of differential operators on O is D(O) = D(Cl(O)) where the closure Cl(O) is a singular cone in g. See [J] and [B-K1] for some results on the geometry and quantization(More)