53 research outputs found
On natural and conformally equivariant quantizations
The concept of conformally equivariant quantizations was introduced by Duval,
Lecomte and Ovsienko in \cite{DLO} for manifolds endowed with flat conformal
structures. They obtained results of existence and uniqueness (up to
normalization) of such a quantization procedure. A natural generalization of
this concept is to seek for a quantization procedure, over a manifold , that
depends on a pseudo-Riemannian metric, is natural and is invariant with respect
to a conformal change of the metric. The existence of such a procedure was
conjectured by P. Lecomte in \cite{Leconj} and proved by C. Duval and V.
Ovsienko in \cite{DO1} for symbols of degree at most 2 and by S. Loubon Djounga
in \cite{Loubon} for symbols of degree 3. In two recent papers \cite{MR,MR1},
we investigated the question of existence of projectively equivariant
quantizations using the framework of Cartan connections. Here we will show how
the formalism developed in these works adapts in order to deal with the
conformally equivariant quantization for symbols of degree at most 3. This will
allow us to easily recover the results of \cite{DO1} and \cite{Loubon}. We will
then show how it can be modified in order to prove the existence of conformally
equivariant quantizations for symbols of degree 4.Comment: 19 page
Operational Entanglement Families of Symmetric Mixed N-Qubit States
We introduce an operational entanglement classification of symmetric mixed
states for an arbitrary number of qubits based on stochastic local operations
assisted with classical communication (SLOCC operations). We define families of
SLOCC entanglement classes successively embedded into each other, we prove that
they are of non-zero measure, and we construct witness operators to distinguish
them. Moreover, we discuss how arbitrary symmetric mixed states can be realized
in the lab via a one-to-one correspondence between well-defined sets of
controllable parameters and the corresponding entanglement families.Comment: 6 pages, 2 figures, published version, Phys. Rev. A, in pres
Equivariant symbol calculus for differential operators acting on forms
We prove the existence and uniqueness of a projectively equivariant symbol
map (in the sense of Lecomte and Ovsienko) for the spaces of differential
operators transforming p-forms into functions. These results hold over a smooth
manifold endowed with a flat projective structure.
As an application, we classify the Vect(M)-equivariant maps from to
over any manifold M, recovering and improving earlier results by N.
Poncin. This provides the complete answer to a question raised by P. Lecomte
about the extension of a certain intrinsic homotopy operator.Comment: 14 page
Projectively equivariant quantizations over the superspace
We investigate the concept of projectively equivariant quantization in the
framework of super projective geometry. When the projective superalgebra
pgl(p+1|q) is simple, our result is similar to the classical one in the purely
even case: we prove the existence and uniqueness of the quantization except in
some critical situations. When the projective superalgebra is not simple (i.e.
in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a
one-parameter family of equivariant quantizations. We also provide explicit
formulas in terms of a generalized divergence operator acting on supersymmetric
tensor fields.Comment: 19 page
Decomposition of symmetric tensor fields in the presence of a flat contact projective structure
Let be an odd-dimensional Euclidean space endowed with a contact 1-form
. We investigate the space of symmetric contravariant tensor fields on
as a module over the Lie algebra of contact vector fields, i.e. over the
Lie subalgebra made up by those vector fields that preserve the contact
structure. If we consider symmetric tensor fields with coefficients in tensor
densities, the vertical cotangent lift of contact form is a contact
invariant operator. We also extend the classical contact Hamiltonian to the
space of symmetric density valued tensor fields. This generalized Hamiltonian
operator on the symbol space is invariant with respect to the action of the
projective contact algebra . The preceding invariant operators lead
to a decomposition of the symbol space (expect for some critical density
weights), which generalizes a splitting proposed by V. Ovsienko
Entanglement Equivalence of -qubit Symmetric States
We study the interconversion of multipartite symmetric -qubit states under
stochastic local operations and classical communication (SLOCC). We demonstrate
that if two symmetric states can be connected with a nonsymmetric invertible
local operation (ILO), then they belong necessarily to the separable, W, or GHZ
entanglement class, establishing a practical method of discriminating subsets
of entanglement classes. Furthermore, we prove that there always exists a
symmetric ILO connecting any pair of symmetric -qubit states equivalent
under SLOCC, simplifying the requirements for experimental implementations of
local interconversion of those states.Comment: Minor correction
Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra
The main contribution of this paper is the construction of a strong duality for the varieties generated by a set of
subalgebras of a semi-primal algebra. We also obtain an axiomatization of the objects of the dual category and develop some algebraic consequences (description of the dual of the finite structures
and algebras, construction of finitely generated free algebras,. . . ).
Eventually, we illustrate this work for the finitely generated varieties of MV-algebras
Natural and projectively equivariant quantizations by means of Cartan Connections
The existence of a natural and projectively equivariant quantization in the
sense of Lecomte [20] was proved recently by M. Bordemann [4], using the
framework of Thomas-Whitehead connections. We give a new proof of existence
using the notion of Cartan projective connections and we obtain an explicit
formula in terms of these connections. Our method yields the existence of a
projectively equivariant quantization if and only if an \sl(m+1,\R)-equivariant
quantization exists in the flat situation in the sense of [18], thus solving
one of the problems left open by M. Bordemann.Comment: 13 page
A classification of bisymmetric polynomial functions over integral domains of characteristic zero
We describe the class of n-variable polynomial functions that satisfy
Acz\'el's bisymmetry property over an arbitrary integral domain of
characteristic zero with identity
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