196 research outputs found
Classicality in discrete Wigner functions
Gibbons et al. [Phys. Rev. A 70, 062101(2004)] have recently defined a class
of discrete Wigner functions W to represent quantum states in a Hilbert space
with finite dimension. We show that the only pure states having non-negative W
for all such functions are stabilizer states, as conjectured by one of us
[Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving
non-negativity of W for all definitions of W form a subgroup of the Clifford
group. This means pure states with non-negative W and their associated unitary
dynamics are classical in the sense of admitting an efficient classical
simulation scheme using the stabilizer formalism.Comment: 10 pages, 1 figur
Some families of density matrices for which separability is easily tested
We reconsider density matrices of graphs as defined in [quant-ph/0406165].
The density matrix of a graph is the combinatorial laplacian of the graph
normalized to have unit trace. We describe a simple combinatorial condition
(the "degree condition") to test separability of density matrices of graphs.
The condition is directly related to the PPT-criterion. We prove that the
degree condition is necessary for separability and we conjecture that it is
also sufficient. We prove special cases of the conjecture involving nearest
point graphs and perfect matchings. We observe that the degree condition
appears to have value beyond density matrices of graphs. In fact, we point out
that circulant density matrices and other matrices constructed from groups
always satisfy the condition and indeed are separable with respect to any
split. The paper isolates a number of problems and delineates further
generalizations.Comment: 14 pages, 4 figure
Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)
Trace formulae for d-regular graphs are derived and used to express the
spectral density in terms of the periodic walks on the graphs under
consideration. The trace formulae depend on a parameter w which can be tuned
continuously to assign different weights to different periodic orbit
contributions. At the special value w=1, the only periodic orbits which
contribute are the non back- scattering orbits, and the smooth part in the
trace formula coincides with the Kesten-McKay expression. As w deviates from
unity, non vanishing weights are assigned to the periodic walks with
back-scatter, and the smooth part is modified in a consistent way. The trace
formulae presented here are the tools to be used in the second paper in this
sequence, for showing the connection between the spectral properties of
d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure
Traces on the Sklyanin algebra and correlation functions of the eight-vertex model
We propose a conjectural formula for correlation functions of the Z-invariant
(inhomogeneous) eight-vertex model. We refer to this conjecture as Ansatz. It
states that correlation functions are linear combinations of products of three
transcendental functions, with theta functions and derivatives as coefficients.
The transcendental functions are essentially logarithmic derivatives of the
partition function per site. The coefficients are given in terms of a linear
functional on the Sklyanin algebra, which interpolates the usual trace on
finite dimensional representations. We establish the existence of the
functional and discuss the connection to the geometry of the classical limit.
We also conjecture that the Ansatz satisfies the reduced qKZ equation. As a
non-trivial example of the Ansatz, we present a new formula for the
next-nearest neighbor correlation functions.Comment: 35 pages, 2 figures, final versio
On a q-extension of Mehta's eigenvectors of the finite Fourier transform for q a root of unity
It is shown that the continuous q-Hermite polynomials for q a root of unity
have simple transformation properties with respect to the classical Fourier
transform. This result is then used to construct q-extended eigenvectors of the
finite Fourier transform in terms of these polynomials.Comment: 12 pages, thoroughly rewritten, the q-extended eigenvectors now
N-periodic with q an M-th root of
Polyhedral Cosmic Strings
Quantum field theory is discussed in M\"obius corner kaleidoscopes using the
method of images. The vacuum average of the stress-energy tensor of a free
field is derived and is shown to be a simple sum of straight cosmic string
expressions, the strings running along the edges of the corners. It does not
seem possible to set up a spin-half theory easily.Comment: 15 pages, 4 text figures not include
Resolution of the Nested Hierarchy for Rational sl(n) Models
We construct Drinfel'd twists for the rational sl(n) XXX-model giving rise to
a completely symmetric representation of the monodromy matrix. We obtain a
polarization free representation of the pseudoparticle creation operators
figuring in the construction of the Bethe vectors within the framework of the
quantum inverse scattering method. This representation enables us to resolve
the hierarchy of the nested Bethe ansatz for the sl(n) invariant rational
Heisenberg model. Our results generalize the findings of Maillet and Sanchez de
Santos for sl(2) models.Comment: 25 pages, no figure
Form factor expansion for thermal correlators
We consider finite temperature correlation functions in massive integrable
Quantum Field Theory. Using a regularization by putting the system in finite
volume, we develop a novel approach (based on multi-dimensional residues) to
the form factor expansion for thermal correlators. The first few terms are
obtained explicitly in theories with diagonal scattering. We also discuss the
validity of the LeClair-Mussardo proposal.Comment: 41 pages; v2: minor corrections, v3: minor correction
Distribution of Eigenvalues for the Modular Group
The two-point correlation function of energy levels for free motion on the
modular domain, both with periodic and Dirichlet boundary conditions, are
explicitly computed using a generalization of the Hardy-Littlewood method. It
is shown that ion the limit of small separations they show an uncorrelated
behaviour and agree with the Poisson distribution but they have prominent
number-theoretical oscillations at larger scale. The results agree well with
numerical simulations.Comment: 72 pages, Latex, the fiogures mentioned in the text are not vital,
but can be obtained upon request from the first Autho
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