8 research outputs found
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,\C)^{\times 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.Comment: 14 pages, IOP macros; slightly expanded versio
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 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
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
An algebraic classification of entangled states
We provide a classification of entangled states that uses new discrete
entanglement invariants. The invariants are defined by algebraic properties of
linear maps associated with the states. We prove a theorem on a correspondence
between the invariants and sets of equivalent classes of entangled states. The
new method works for an arbitrary finite number of finite-dimensional state
subspaces. As an application of the method, we considered a large selection of
cases of three subspaces of various dimensions. We also obtain an entanglement
classification of four qubits, where we find 27 fundamental sets of classes.Comment: published versio