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 ${\bf CP}^3$. 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 $n$-qubit ones related to Grassmannians of $l$-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 ${\bf CP}^3$ furnished with a fixed quadric.Comment: 19 page

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

Quelques remarques sur une épidémie de grippe équine qui sévit en décembre 1965 à l’annexe de l’Institut Pasteur de Carches

Au début du mois de décembre 1965 une épidémie de grippe équine a éclaté dans les écuries de l’Institut Pasteur de Carches où 22 p. 100 de l’effectif furent atteints. Le taux de mortalité fut de 4 p. 100. Dix souches furent isolées ; l’étude antigénique de ces souches fut effectuée comparativement aux diverses souches équines antérieurement isolées en Europe et en Amérique. Un vaccin quadrivalent a été préparé, il est en cours d’expérimen tation

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

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

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

Superelastic Behavior of Biomedical Metallic Alloys

In this this work, superelastic NiTi and Ni-free Ti-23Hf-3Mo-4Sn biomedical alloys were investigated by tensile tests in relationship with their microstructures. To follow the stress-induced martensitic transformations occurring in these alloys, in situ tensile tests under synchrotron beam were conducted. In NiTi, an intermediate trigonal R phase, which is first stress-induced before the B19 ' martensitic phase, was identified. However, the Ti-23Hf-3Mo-4Sn alloy does not present a transitional phase, and a direct beta into alpha '' reversible stress-induced martensitic transformation was observed. With NiTi, all the applied strain is recovered after unloading, and no residual plastic deformation occurs. However, the strain is not completely recovered with the Ti-23Hf-3Mo-4Sn alloy, and residual plastic strain was observed to prevent a complete recovery, thus explaining why the strain recovery is lower for Ti-23Hf-3Mo-4Sn compared with NiTi. We also showed that the maximum strain recovery depends on the texture in the Ti-23Hf-3Mo-4Sn alloy. The favorable texture leading to the highest strain recovery (4.6 pct) is the {111}< 110 and rang;(beta) texture, which can be obtained by a short-time solution treatment (0.3 ks) at 1073 K with this alloy

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

Quadratic pseudosupersymmetry in two-level systems

Using the intertwining relation we construct a pseudosuperpartner for a (non-Hermitian) Dirac-like Hamiltonian describing a two-level system interacting in the rotating wave approximation with the electric component of an electromagnetic field. The two pseudosuperpartners and pseudosupersymmetry generators close a quadratic pseudosuperalgebra. A class of time dependent electric fields for which the equation of motion for a two level system placed in this field can be solved exactly is obtained. New interesting phenomenon is observed. There exists such a time-dependent detuning of the field frequency from the resonance value that the probability to populate the excited level ceases to oscillate and becomes a monotonically growing function of time tending to 3/4. It is shown that near this fixed excitation regime the probability exhibits two kinds of oscillations. The oscillations with a small amplitude and a frequency close to the Rabi frequency (fast oscillations) take place at the background of the ones with a big amplitude and a small frequency (slow oscillations). During the period of slow oscillations the minimal value of the probability to populate the excited level may exceed 1/2 suggesting for an ensemble of such two-level atoms the possibility to acquire the inverse population and exhibit lasing properties.Comment: 5 figure