Jean-Gabriel Luque

Polynomial invariants of four qubits

We describe explicitly the algebra of polynomial functions on the Hilbert space of four-qubit states that are invariant under the group SL(2,C){sup 4} of stochastic local quantum operations assisted by classical communication. From this description, we obtain a closed formula for the hyperdeterminant in terms of low degree invariants.

A complete set of covariants of the four qubit system

We obtain a complete and minimal set of 170 generators for the algebra of SL(2, )×4 covariants of a binary quadrilinear form. Interpreted in terms of a four qubit system, this describes in particular the algebraic varieties formed by the orbits of local filtering operations in its projective Hilbert space. Also, this sheds some light on the local unitary invariants, and provides all the possible building blocks for the construction of entanglement measures for such a system.

Hankel hyperdeterminants and Selberg integrals

We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form Ai1i2...ik = f(i1 + i2 + ... + ik)). It is found that many classical properties of Hankel determinants can be generalized, and a connection with Selberg type integrals is established. In particular, Selbergs original formula amounts to the evaluation of all Hankel hyperdeterminants built from the moments of the Jacobi polynomials. Many higher dimensional analogues of classical Hankel determinants are evaluated in closed form. The Toeplitz case is also briefly discussed. In physical terms, both cases are related to the partition functions of one-dimensional Coulomb systems with logarithmic potential.

The moduli space of three-qutrit states

We study the invariant theory of trilinear forms over a three-dimensional complex vector space, and apply it to investigate the behavior of pure entangled three-partite qutrit states and their normal forms under local filtering operations (SLOCC). We describe the orbit space of the SLOCC group SL(3,C)×3 both in its affine and projective versions in terms of a very symmetric normal form parametrized by three complex numbers. The parameters of the possible normal forms of a given state are roots of an algebraic equation, which is proved to be solvable by radicals. The structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes.

Pfaffian and Hafnian identities in shuffle algebras

Chens lemma on iterated integrals implies that certain identities involving multiple integrals, such as the de Bruijn and Wick formulas, amount to combinatorial identities for Pfaffians and Hafnians in shuffle algebras. We provide direct algebraic proofs of such shuffle identities, and obtain various generalizations. We also discuss some Pfaffian identities due to Sundquist and Ishikawa-Wakayama, and a Cauchy formula for anticommutative symmetric functions. Finally, we extend some of the previous considerations to hyper-Pfaffians and hyper-Hafnians.

Direct and dual laws for automata with multiplicities

We present here theoretical results coming from the implementation of the package called AMULT (automata with multiplicities). We show that classical formulas are optimal for the bounds. Especially they are almost everywhere optimal for the fields R and C. We characterize the dual laws preserving rationality and examine compatibility between the geometry of the K-automata andthese laws. Copyright 2001 Elsevier Science B.V.

Unitary invariants of qubit systems

We give an algorithm allowing the construction of bases of local unitary invariants of pure k-qubit states from a knowledge of the polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are made explicit and compared with various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.

Hyperdeterminantal calculations of Selberg's and Aomoto's integrals

The hyperdeteminants considered here are the simplest analogues of determinants for higher rank tensors which have been defined by Cayley, and apply only to tensors with an even number of indices. It was shown in a previous article that the calculation of certain multidimensional integrals could be reduced to the evaluation of hyperdeterminants of Hankel type. Here, this computation is carried out by purely algebraic means in the cases of Selbergs and Aomotos integrals.

Geometric descriptions of entangled states by auxiliary varieties

The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting, we describe well-known classifications of multipartite entanglement such as 2 × 2 × (n + 1), for n ⩾ 1, quantum systems and a new description with the 2 × 3 × 3 quantum system. Our results complete the approach of Miyake and make stronger connections with recent work of algebraic geometers. Moreover, for the quantum systems detailed in this paper, we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.

Highest weight Macdonald and Jack polynomials

Fractional quantum Hall states of particles in the lowest Landau levels are described by multivariate polynomials. The incompressible liquid states when described on a sphere are fully invariant under the rotation group. Excited quasiparticle/quasihole states are members of multiplets under the rotation group and generically there is a nontrivial highest weight member of the multiplet from which all states can be constructed. Some of the trial states proposed in the literature belong to classical families of symmetric polynomials. In this paper we study Macdonald and Jack polynomials that are highest weight states. For Macdonald polynomials, it is a (q, t)-deformation of the raising angular momentum operator that defines the highest weight condition. By specialization of the parameters we obtain a classification of the highest weight Jack polynomials. Our results are valid in the case of staircase and rectangular partition indexing the polynomials.