The twistor geometry of three-qubit entanglement

A geometrical description of three qubit entanglement is given. A part of the transformations corresponding to stochastic local operations and classical communication on the qubits is regarded as a gauge degree of freedom. Entangled states can be represented by the points of the Klein quadric ${\cal Q}$ a space known from twistor theory. It is shown that three-qubit invariants are vanishing on special subspaces of ${\cal Q}$. An invariant vanishing for the $GHZ$ class is proposed. A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.Comment: 4 pages RevTeX

Where are the Hedgehogs in Nematics?

In experiments which take a liquid crystal rapidly from the isotropic to the nematic phase, a dense tangle of defects is formed. In nematics, there are in principle both line and point defects (``hedgehogs''), but no point defects are observed until the defect network has coarsened appreciably. In this letter the expected density of point defects is shown to be extremely low, approximately $10^{-8}$ per initially correlated domain, as result of the topology (specifically, the homology) of the order parameter space.Comment: 6 pages, latex, 1 figure (self-unpacking PostScript)

Fermion Quasi-Spherical Harmonics

Spherical Harmonics, $Y_\ell^m(\theta,\phi)$, are derived and presented (in a Table) for half-odd-integer values of $\ell$ and $m$. These functions are eigenfunctions of $L^2$ and $L_z$ written as differential operators in the spherical-polar angles, $\theta$ and $\phi$. The Fermion Spherical Harmonics are a new, scalar and angular-coordinate-dependent representation of fermion spin angular momentum. They have $4\pi$ symmetry in the angle $\phi$, and hence are not single-valued functions on the Euclidean unit sphere; they are double-valued functions on the sphere, or alternatively are interpreted as having a double-sphere as their domain.Comment: 16 pages, 2 Tables. Submitted to J.Phys.

Quantum strategies

We consider game theory from the perspective of quantum algorithms. Strategies in classical game theory are either pure (deterministic) or mixed (probabilistic). We introduce these basic ideas in the context of a simple example, closely related to the traditional Matching Pennies game. While not every two-person zero-sum finite game has an equilibrium in the set of pure strategies, von Neumann showed that there is always an equilibrium at which each player follows a mixed strategy. A mixed strategy deviating from the equilibrium strategy cannot increase a player's expected payoff. We show, however, that in our example a player who implements a quantum strategy can increase his expected payoff, and explain the relation to efficient quantum algorithms. We prove that in general a quantum strategy is always at least as good as a classical one, and furthermore that when both players use quantum strategies there need not be any equilibrium, but if both are allowed mixed quantum strategies there must be.Comment: 8 pages, plain TeX, 1 figur

Global embedding of the Kerr black hole event horizon into hyperbolic 3-space

An explicit global and unique isometric embedding into hyperbolic 3-space, H^3, of an axi-symmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into H^3 of surfaces of revolution having negative, but finite, Gaussian curvature at smooth fixed points of the U(1) isometry. As an example, we exhibit the global embedding of the Kerr-Newman event horizon into H^3, for arbitrary values of the angular momentum. For this example, considering a quotient of H^3 by the Picard group, we show that the hyperbolic embedding fits in a fundamental domain of the group up to a slightly larger value of the angular momentum than the limit for which a global embedding into Euclidean 3-space is possible. An embedding of the double-Kerr event horizon is also presented, as an example of an embedding which cannot be made global.Comment: 16 pages, 13 figure

Modular Invariance of Finite Size Corrections and a Vortex Critical Phase

We analyze a continuous spin Gaussian model on a toroidal triangular lattice with periods $L_0$ and $L_1$ where the spins carry a representation of the fundamental group of the torus labeled by phases $u_0$ and $u_1$. We find the {\it exact finite size and lattice corrections}, to the partition function $Z$, for arbitrary mass $m$ and phases $u_i$. Summing $Z^{-1/2}$ over phases gives the corresponding result for the Ising model. The limits $m\rightarrow0$ and $u_i\rightarrow0$ do not commute. With $m=0$ the model exhibits a {\it vortex critical phase} when at least one of the $u_i$ is non-zero. In the continuum or scaling limit, for arbitrary $m$, the finite size corrections to $-\ln Z$ are {\it modular invariant} and for the critical phase are given by elliptic theta functions. In the cylinder limit $L_1\rightarrow\infty$ the ``cylinder charge'' $c(u_0,m^2L_0^2)$ is a non-monotonic function of $m$ that ranges from $2(1+6u_0(u_0-1))$ for $m=0$ to zero for $m\rightarrow\infty$.Comment: 12 pages of Plain TeX with two postscript figure insertions called torusfg1.ps and torusfg2.ps which can be obtained upon request from [email protected]

Multiple zero modes of the Dirac operator in three dimensions

One of the key properties of Dirac operators is the possibility of a degeneracy of zero modes. For the Abelian Dirac operator in three dimensions the construction of multiple zero modes has been sucessfully carried out only very recently. Here we generalise these results by discussing a much wider class of Dirac operators together with their zero modes. Further we show that those Dirac operators that do admit zero modes may be related to Hopf maps, where the Hopf index is related to the number of zero modes in a simple way.Comment: Latex file, 20 pages, no figure

Effect of retardation on dynamical mass generation in two-dimensional QED at finite temperature

The effect of retardation on dynamical mass generation in is studied, in the imaginary time formalism. The photon porarization tensor is evaluated to leading order in 1/N (N is the number of flavours), and simple closed form expressions are found for the fully retarded longitudinal and transverse propagators, which have the correct limit when T goes to zero. The resulting S-D equation for the fermion mass (at order 1/N) has an infrared divergence associated with the contribution of the transverse photon propagator; only the longitudinal contribution is retained, as in earlier treatments. For solutions of constant mass, it is found that the retardation reduces the value of the parameter r (the ratio of twice the mass to the critical temperature) from about 10 to about 6. The gap equation is then solved allowing for the mass to depend on frequency. It was found that the r value remained close to 6. Possibilities for including the transverse propagator are discussed.Comment: 26 pages 8 figure

Equations of Motion of Spinning Relativistic Particle in Electromagnetic and Gravitational Fields

We consider the motion of a spinning relativistic particle in external electromagnetic and gravitational fields, to first order in the external field, but to an arbitrary order in spin. The noncovariant spin formalism is crucial for the correct description of the influence of the spin on the particle trajectory. We show that the true coordinate of a relativistic spinning particle is its naive, common coordinate \r. Concrete calculations are performed up to second order in spin included. A simple derivation is presented for the gravitational spin-orbit and spin-spin interactions of a relativistic particle. We discuss the gravimagnetic moment (GM), a specific spin effect in general relativity. It is shown that for the Kerr black hole the gravimagnetic ratio, i.e., the coefficient at the GM, equals unity (just as for the charged Kerr hole the gyromagnetic ratio equals two). The equations of motion obtained for relativistic spinning particle in external gravitational field differ essentially from the Papapetrou equations.Comment: 32 pages, latex, Plenary talk at the Fairbank Meeting on the Lense--Thirring Effect, Rome-Pescara, 29/6-4/7 199