812 research outputs found
Geometry of entangled states, Bloch spheres and Hopf fibrations
We discuss a generalization to 2 qubits of the standard Bloch sphere
representation for a single qubit, in the framework of Hopf fibrations of high
dimensional spheres by lower dimensional spheres. The single qubit Hilbert
space is the 3-dimensional sphere S3. The S2 base space of a suitably oriented
S3 Hopf fibration is nothing but the Bloch sphere, while the circular fibres
represent the qubit overall phase degree of freedom. For the two qubits case,
the Hilbert space is a 7-dimensional sphere S7, which also allows for a Hopf
fibration, with S3 fibres and a S4 base. A main striking result is that
suitably oriented S7 Hopf fibrations are entanglement sensitive. The relation
with the standard Schmidt decomposition is also discussedComment: submitted to J. Phys.
A formula for the number of tilings of an octagon by rhombi
We propose the first algebraic determinantal formula to enumerate tilings of
a centro-symmetric octagon of any size by rhombi. This result uses the
Gessel-Viennot technique and generalizes to any octagon a formula given by
Elnitsky in a special case.Comment: New title. Minor improvements. To appear in Theoretical Computer
Science, special issue on "Combinatorics of the Discrete Plane and Tilings
Entanglement in a second order quantum phase transition
We consider a system of mutually interacting spin 1/2 embedded in a
transverse magnetic field which undergo a second order quantum phase
transition. We analyze the entanglement properties and the spin squeezing of
the ground state and show that, contrarily to the one-dimensional case, a
cusp-like singularity appears at the critical point , in the
thermodynamic limit. We also show that there exists a value above which the ground state is not spin squeezed despite a
nonvanishing concurrence.Comment: 4 pages, 4 EPS figures, minor corrections added and title change
Landau levels in quasicrystals
Two-dimensional tight-binding models for quasicrystals made of plaquettes
with commensurate areas are considered. Their energy spectrum is computed as a
function of an applied perpendicular magnetic field. Landau levels are found to
emerge near band edges in the zero-field limit. Their existence is related to
an effective zero-field dispersion relation valid in the continuum limit. For
quasicrystals studied here, an underlying periodic crystal exists and provides
a natural interpretation to this dispersion relation. In addition to the slope
(effective mass) of Landau levels, we also study their width as a function of
the magnetic flux per plaquette and identify two fundamental broadening
mechanisms: (i) tunneling between closed cyclotron orbits and (ii) individual
energy displacement of states within a Landau level. Interestingly, the typical
broadening of the Landau levels is found to behave algebraically with the
magnetic field with a nonuniversal exponent.Comment: 14 pages, 9 figure
Geometric Approach to Digital Quantum Information
We present geometric methods for uniformly discretizing the continuous
N-qubit Hilbert space. When considered as the vertices of a geometrical figure,
the resulting states form the equivalent of a Platonic solid. The
discretization technique inherently describes a class of pi/2 rotations that
connect neighboring states in the set, i.e. that leave the geometrical figures
invariant. These rotations are shown to generate the Clifford group, a general
group of discrete transformations on N qubits. Discretizing the N-qubit Hilbert
space allows us to define its digital quantum information content, and we show
that this information content grows as N^2. While we believe the discrete sets
are interesting because they allow extra-classical behavior--such as quantum
entanglement and quantum parallelism--to be explored while circumventing the
continuity of Hilbert space, we also show how they may be a useful tool for
problems in traditional quantum computation. We describe in detail the discrete
sets for one and two qubits.Comment: Introduction rewritten; 'Sample Application' section added. To appear
in J. of Quantum Information Processin
Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions
Three-dimensional integer partitions provide a convenient representation of
codimension-one three-dimensional random rhombus tilings. Calculating the
entropy for such a model is a notoriously difficult problem. We apply
transition matrix Monte Carlo simulations to evaluate their entropy with high
precision. We consider both free- and fixed-boundary tilings. Our results
suggest that the ratio of free- and fixed-boundary entropies is
, and can be interpreted as the ratio of the
volumes of two simple, nested, polyhedra. This finding supports a conjecture by
Linde, Moore and Nordahl concerning the ``arctic octahedron phenomenon'' in
three-dimensional random tilings
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