Geometrical approach to SU(2) navigation with Fibonacci anyons

Topological quantum computation with Fibonacci anyons relies on the possibility of efficiently generating unitary transformations upon pseudoparticles braiding. The crucial fact that such set of braids has a dense image in the unitary operations space is well known; in addition, the Solovay-Kitaev algorithm allows to approach a given unitary operation to any desired accuracy. In this paper, the latter task is fulfilled with an alternative method, in the SU(2) case, based on a generalization of the geodesic dome construction to higher dimension.Comment: 12 pages, 5 figure

Generalized quasiperiodic Rauzy tilings

We present a geometrical description of new canonical $d$-dimensional codimension one quasiperiodic tilings based on generalized Fibonacci sequences. These tilings are made up of rhombi in 2d and rhombohedra in 3d as the usual Penrose and icosahedral tilings. Thanks to a natural indexing of the sites according to their local environment, we easily write down, for any approximant, the sites coordinates, the connectivity matrix and we compute the structure factor.Comment: 11 pages, 3 EPS figures, final version with minor change

Correlation induced non-Abelian quantum holonomies

In the context of two-particle interferometry, we construct a parallel transport condition that is based on the maximization of coincidence intensity with respect to local unitary operations on one of the subsystems. The dependence on correlation is investigated and it is found that the holonomy group is generally non-Abelian, but Abelian for uncorrelated systems. It is found that our framework contains the L\'{e}vay geometric phase [2004 {\it J. Phys. A: Math. Gen.} {\bf 37} 1821] in the case of two-qubit systems undergoing local SU(2) evolutions.Comment: Minor corrections; journal reference adde

Interaction induced delocalisation for two particles in a periodic potential

We consider two interacting particles evolving in a one-dimensional periodic structure embedded in a magnetic field. We show that the strong localization induced by the magnetic field for particular values of the flux per unit cell is destroyed as soon as the particles interact. We study the spectral and the dynamical aspects of this transition.Comment: 4 pages, 5 EPS figures, minor misprints correcte

A remark on the trace-map for the Silver mean sequence

In this work we study the Silver mean sequence based on substitution rules by means of a transfer-matrix approach. Using transfer-matrix method we find a recurrence relation for the traces of general transfer-matrices which characterizes electronic properties of the quasicrystal in question. We also find an invariant of the trace-map.Comment: 5 pages, minor improvements in style and presentation of calculation

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

Quantum dynamics in high codimension tilings: from quasiperiodicity to disorder

We analyze the spreading of wavepackets in two-dimensional quasiperiodic and random tilings as a function of their codimension, i.e. of their topological complexity. In the quasiperiodic case, we show that the diffusion exponent that characterizes the propagation decreases when the codimension increases and goes to 1/2 in the high codimension limit. By constrast, the exponent for the random tilings is independent of their codimension and also equals 1/2. This shows that, in high codimension, the quasiperiodicity is irrelevant and that the topological disorder leads in every case, to a diffusive regime, at least in the time scale investigated here.Comment: 4 pages, 5 EPS figure

Geometry of the 3-Qubit State, Entanglement and Division Algebras

We present a generalization to 3-qubits of the standard Bloch sphere representation for a single qubit and of the 7-dimensional sphere representation for 2 qubits presented in Mosseri {\it et al.}\cite{Mosseri2001}. The Hilbert space of the 3-qubit system is the 15-dimensional sphere $S^{15}$, which allows for a natural (last) Hopf fibration with $S^8$ as base and $S^7$ as fiber. A striking feature is, as in the case of 1 and 2 qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling 3 qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the 3-qubit state. Conjectures on the possibility to generalize the construction for higher qubit states are also discussed.Comment: 12 pages, 2 figures, final versio

The geometry of entanglement: metrics, connections and the geometric phase

Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.Comment: 42 page