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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)

- RANEE BRYLINSKI
- 2000

Lecomte and Ovsienko constructed SL n+1 (R)-equivariant quantization maps Q λ for symbols of differential operators on λ-densities on RP n. We derive some formulas for the associated graded equivariant star products ⋆ λ on the symbol algebra Pol(T * RP n). These give some measure of the failure of locality. Our main result expresses (for n odd) the… (More)

- Ranee Brylinski, Bertram Kostant
- 1992

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)

- Ranee Brylinski
- 1998

In this paper, we begin a quantization program for nilpotent orbits O R of a real semisimple Lie group G R. These orbits arise naturally as the coadjoint orbits of G R which are stable under scaling, and thus they have a canonical symplectic structure ω where the G R-action is Hamiltonian. These orbits and their covers generalize the oscillator phase space… (More)

- ALEXANDER ASTASHKEVICH, RANEE BRYLINSKI
- 2000

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)

- RANEE BRYLINSKI
- 2001

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)

- Alexander Astashkevich, Ranee Brylinski
- 1998

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)

- Jean-Luc Brylinski, Ranee Brylinski
- ArXiv
- 2003

Immanants are polynomial functions of n by n matrices attached to irre-ducible characters of the symmetric group Sn, or equivalently to Young diagrams of size n. Immanants include determinants and permanents as extreme cases. Valiant proved that computation of permanents is a complete problem in his algebraic model of NP theory , i.e., it is VNP-complete.… (More)

We study the polynomial functions on tensor states in (C n) ⊗k which are invariant under SU (n) k. We describe the space of invariant polynomials in terms of symmetric group representations. For k even, the smallest degree for invariant polynomials is n and in degree n we find a natural generalization of the determinant. For n, d fixed, we describe the… (More)

The first obstacle in building a Geometric Quantization theory for nilpotent orbits of a real semisimple Lie group has been the lack of an invariant polarization. In order to generalize the Fock space construction of the quantum mechanical oscillator, a polarization of the symplectic orbit invariant under the maximal compact subgroup is required. In this… (More)