1,454 research outputs found
Localization of the Grover walks on spidernets and free Meixner laws
A spidernet is a graph obtained by adding large cycles to an almost regular
tree and considered as an example having intermediate properties of lattices
and trees in the study of discrete-time quantum walks on graphs. We introduce
the Grover walk on a spidernet and its one-dimensional reduction. We derive an
integral representation of the -step transition amplitude in terms of the
free Meixner law which appears as the spectral distribution. As an application
we determine the class of spidernets which exhibit localization. Our method is
based on quantum probabilistic spectral analysis of graphs.Comment: 32 page
Monotone independence, comb graphs and Bose-Einstein condensation
The adjacency matrix of a comb graph is decomposed into a sum of monotone independent random variables with respect to the vacuum state. The vacuum spectral distribution is shown to be asymptotically the arcsine law as a consequence of the monotone central limit theorem. As an example the comb lattice is studied with explicit calculation
On the Lichnerowicz conjecture for CR manifolds with mixed signature
We construct examples of nondegenerate CR manifolds with Levi form of
signature , , which are compact, not locally CR flat, and
admit essential CR vector fields. We also construct an example of a noncompact
nondegenerate CR manifold with signature which is not locally CR flat
and admits an essential CR vector fields. These provide counterexamples to the
analogue of the Lichnerowicz conjecture for CR manifolds with mixed signature.Comment: 7 page
Wigner formula of rotation matrices and quantum walks
Quantization of a random-walk model is performed by giving a qudit (a
multi-component wave function) to a walker at site and by introducing a quantum
coin, which is a matrix representation of a unitary transformation. In quantum
walks, the qudit of walker is mixed according to the quantum coin at each time
step, when the walker hops to other sites. As special cases of the quantum
walks driven by high-dimensional quantum coins generally studied by Brun,
Carteret, and Ambainis, we study the models obtained by choosing rotation as
the unitary transformation, whose matrix representations determine quantum
coins. We show that Wigner's -dimensional unitary representations of
rotations with half-integers 's are useful to analyze the probability laws
of quantum walks. For any value of half-integer , convergence of all moments
of walker's pseudovelocity in the long-time limit is proved. It is generally
shown for the present models that, if is even, the probability measure
of limit distribution is given by a superposition of terms of scaled
Konno's density functions, and if is odd, it is a superposition of
terms of scaled Konno's density functions and a Dirac's delta function at the
origin. For the two-, three-, and four-component models, the probability
densities of limit distributions are explicitly calculated and their dependence
on the parameters of quantum coins and on the initial qudit of walker is
completely determined. Comparison with computer simulation results is also
shown.Comment: v2: REVTeX4, 15 pages, 4 figure
The Momentum Constraints of General Relativity and Spatial Conformal Isometries
Transverse-tracefree (TT-) tensors on , with an
asymptotically flat metric of fast decay at infinity, are studied. When the
source tensor from which these TT tensors are constructed has fast fall-off at
infinity, TT tensors allow a multipole-type expansion. When has no
conformal Killing vectors (CKV's) it is proven that any finite but otherwise
arbitrary set of moments can be realized by a suitable TT tensor. When CKV's
exist there are obstructions -- certain (combinations of) moments have to
vanish -- which we study.Comment: 16 page
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