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

Black holes admitting a Freudenthal dual

The quantised charges x of four dimensional stringy black holes may be assigned to elements of an integral Freudenthal triple system whose automorphism group is the corresponding U-duality and whose U-invariant quartic norm Delta(x) determines the lowest order entropy. Here we introduce a Freudenthal duality x -> \tilde{x}, for which \tilde{\tilde{x}}=-x. Although distinct from U-duality it nevertheless leaves Delta(x) invariant. However, the requirement that \tilde{x} be integer restricts us to the subset of black holes for which Delta(x) is necessarily a perfect square. The issue of higher-order corrections remains open as some, but not all, of the discrete U-duality invariants are Freudenthal invariant. Similarly, the quantised charges A of five dimensional black holes and strings may be assigned to elements of an integral Jordan algebra, whose cubic norm N(A) determines the lowest order entropy. We introduce an analogous Jordan dual A*, with N(A) necessarily a perfect cube, for which A**=A and which leaves N(A) invariant. The two dualities are related by a 4D/5D lift.Comment: 32 pages revtex, 10 tables; minor corrections, references adde

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

Freudenthal triple classification of three-qubit entanglement

We show that the three-qubit entanglement classes: (0) Null, (1) Separable A-B-C, (2a) Biseparable A-BC, (2b) Biseparable B-CA, (2c) Biseparable C-AB, (3) W and (4) GHZ correspond respectively to ranks 0, 1, 2a, 2b, 2c, 3 and 4 of a Freudenthal triple system defined over the Jordan algebra C+C+C. We also compute the corresponding SLOCC orbits.Comment: 11 pages, 2 figures, 6 tables, revtex; minor corrections, references added; version appearing in Phys. Rev.

The square of the Vandermonde determinant and its q-generalisation

The Vandermonde determinant plays a crucial role in the quantum Hall effect via Laughlin's wavefunction ansatz. Herein the properties of the square of the Vandermonde determinant as a symmetric function are explored in detail. Important properties satisfied by the coefficients arising in the expansion of the square of the Vandermonde determinant in terms of Schur functions are developed and generalized to q-dependent coefficients via the q-discriminant. Algorithms for the efficient calculation of the q-dependent coefficients as finite polynomials in q are developed. The properties, such as the factorization of the q-dependent coefficients, are exposed. Further light is shed upon the vanishing of certain expansion coefficients at q = 1. The q-generalization of the sum rule for the squares of the coefficients is derived. A number of compelling conjectures are stated

Products and symmetrized powers of irreducible representations of SO*(2n)

The calculation of branching rules, tensor products and plethysms of the infinite-dimensional harmonic series unitary irreducible representations of the non-compact group is considered and the duality between and Sp(2k) exploited. The branching rule for the restriction of an arbitrary harmonic series irreducible representation of to U(n) is derived, and the decomposition is given explicitly for each of the infinite number of fundamental harmonic series irreducible representations, , of whose direct sum constitutes the metaplectic representation, H, of . A concise expression for the decomposition of tensor products is derived and a complete analysis of the terms in both and is given. A general formula for plethysms of arbitrary irreducible representations of is derived and its implementation illustrated both by means of a detailed generic example and by a complete determination of the symmetric and antisymmetric terms of . Finally, relationships that arise from the embedding of the product groups and in the metaplectic group Mp(4nk) are discussed

Kostka Numbers and Littlewood-Richardson Coefficients: Distributed Computation

International audienc

A finite subgroup of the exceptional Lie group G2

With a view to further refining the use of the exceptional group G2 in atomic and nuclear spectroscopy, it is confirmed that a simple finite subgroup L168~PSL2(7) of order 168 of the symmetric group S8 is also a subgroup of G2. It is established by character theoretic and other methods that there are two distinct embeddings of L168 in G2, analogous to the two distinct embeddings of SO(3) in G2. Relevant branching rules, tensor products and symmetrized tensor products are tabulated. As a stimulus to further applications the branching rules are given for the restriction from L168 to the octahedral crystallographic point group O