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

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THIBON A., TARDIEU C., CAMOIN A. 2017

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 ${\bf C}^{2^N}\simeq{\bf C}^L\otimes{\bf C}^l$ with $L=2^{N-n}\geq l=2^n$. Such partitions can be given a nice geometrical interpretation in terms of Grassmannians Gr(L,l) of l-planes in ${\bf C}^L$ 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

Algebraic invariants of five qubits

The Hilbert series of the algebra of polynomial invariants of pure states of five qubits is obtained, and the simplest invariants are computed.Comment: 4 pages, revtex. Short discussion of quant-ph/0506073 include

Crystal Graphs and qq-Analogues of Weight Multiplicities for the Root System AnA_n

We give an expression of the $q$-analogues of the multiplicities of weights in irreducible \sl_{n+1}-modules in terms of the geometry of the crystal graph attached to the corresponding U_q(\sl_{n+1})-modules. As an application, we describe multivariate polynomial analogues of the multiplicities of the zero weight, refining Kostant's generalized exponents.Comment: LaTeX file with epic, eepic pictures, 17 pages, November 1994, to appear in Lett. Math. Phy