182 research outputs found
On dual canonical bases
The dual basis of the canonical basis of the modified quantized enveloping
algebra is studied, in particular for type . The construction of a basis for
the coordinate algebra of the quantum matrices is appropriate for
the study the multiplicative property. It is shown that this basis is invariant
under multiplication by certain quantum minors including the quantum
determinant. Then a basis of quantum SL(n) is obtained by setting the quantum
determinant to one. This basis turns out to be equivalent to the dual canonical
basis
Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)
We investigate the geometry of the four qubit systems by means of algebraic
geometry and invariant theory, which allows us to interpret certain entangled
states as algebraic varieties. More precisely we describe the nullcone, i.e.,
the set of states annihilated by all invariant polynomials, and also the so
called third secant variety, which can be interpreted as the generalization of
GHZ-states for more than three qubits. All our geometric descriptions go along
with algorithms which allow us to identify any given state in the nullcone or
in the third secant variety as a point of one of the 47 varieties described in
the paper. These 47 varieties correspond to 47 non-equivalent entanglement
patterns, which reduce to 15 different classes if we allow permutations of the
qubits.Comment: 48 pages, 7 tables, 13 figures, references and remarks added (v2
On the geometry of a class of N-qubit entanglement monotones
A family of N-qubit entanglement monotones invariant under stochastic local
operations and classical communication (SLOCC) is defined. This class of
entanglement monotones includes the well-known examples of the concurrence, the
three-tangle, and some of the four, five and N-qubit SLOCC invariants
introduced recently. The construction of these invariants is based on bipartite
partitions of the Hilbert space in the form with . Such partitions can be given
a nice geometrical interpretation in terms of Grassmannians Gr(L,l) of l-planes
in that can be realized as the zero locus of quadratic polinomials
in the complex projective space of suitable dimension via the Plucker
embedding. The invariants are neatly expressed in terms of the Plucker
coordinates of the Grassmannian.Comment: 7 pages RevTex, Submitted to Physical Review
New invariants for entangled states
We propose new algebraic invariants that distinguish and classify entangled
states. Considering qubits as well as higher spin systems, we obtained complete
entanglement classifications for cases that were either unsolved or only
conjectured in the literature.Comment: published versio
On the geometry of four qubit invariants
The geometry of four-qubit entanglement is investigated. We replace some of
the polynomial invariants for four-qubits introduced recently by new ones of
direct geometrical meaning. It is shown that these invariants describe four
points, six lines and four planes in complex projective space . For
the generic entanglement class of stochastic local operations and classical
communication they take a very simple form related to the elementary symmetric
polynomials in four complex variables. Moreover, their magnitudes are
entanglement monotones that fit nicely into the geometric set of -qubit ones
related to Grassmannians of -planes found recently. We also show that in
terms of these invariants the hyperdeterminant of order 24 in the four-qubit
amplitudes takes a more instructive form than the previously published
expressions available in the literature. Finally in order to understand two,
three and four-qubit entanglement in geometric terms we propose a unified
setting based on furnished with a fixed quadric.Comment: 19 page
A Hopf laboratory for symmetric functions
An analysis of symmetric function theory is given from the perspective of the
underlying Hopf and bi-algebraic structures. These are presented explicitly in
terms of standard symmetric function operations. Particular attention is
focussed on Laplace pairing, Sweedler cohomology for 1- and 2-cochains, and
twisted products (Rota cliffordizations) induced by branching operators in the
symmetric function context. The latter are shown to include the algebras of
symmetric functions of orthogonal and symplectic type. A commentary on related
issues in the combinatorial approach to quantum field theory is given.Comment: 29 pages, LaTeX, uses amsmat
Wigner transform and pseudodifferential operators on symmetric spaces of non-compact type
We obtain a general expression for a Wigner transform (Wigner function) on
symmetric spaces of non-compact type and study the Weyl calculus of
pseudodifferential operators on them
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