216 research outputs found
Noiseless Quantum Circuits for the Peres Separability Criterion
In this Letter we give a method for constructing sets of simple circuits that
can determine the spectrum of a partially transposed density matrix, without
requiring either a tomographically complete POVM or the addition of noise to
make the spectrum non-negative. These circuits depend only on the dimension of
the Hilbert space and are otherwise independent of the state.Comment: 4 pages RevTeX, 7 figures encapsulated postscript. v5: title changed
slightly, more-or-less equivalent to the published versio
Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras
This is a continuation of our "Lecture on Kac--Moody Lie algebras of the
arithmetic type" \cite{25}.
We consider hyperbolic (i.e. signature ) integral symmetric bilinear
form (i.e. hyperbolic lattice), reflection group
, fundamental polyhedron \Cal M of and an acceptable
(corresponding to twisting coefficients) set P({\Cal M})\subset M of vectors
orthogonal to faces of \Cal M (simple roots). One can construct the
corresponding Lorentzian Kac--Moody Lie algebra {\goth g}={\goth
g}^{\prime\prime}(A(S,W,P({\Cal M}))) which is graded by .
We show that \goth g has good behavior of imaginary roots, its denominator
formula is defined in a natural domain and has good automorphic properties if
and only if \goth g has so called {\it restricted arithmetic type}. We show
that every finitely generated (i.e. P({\Cal M}) is finite) algebra {\goth
g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1))) may be embedded to {\goth
g}^{\prime\prime}(A(S,W,P({\Cal M}))) of the restricted arithmetic type. Thus,
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a
natural class to study.
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the
best automorphic properties for the denominator function if they have {\it a
lattice Weyl vector }. Lorentzian Kac--Moody Lie algebras of the
restricted arithmetic type with generalized lattice Weyl vector are
called {\it elliptic}Comment: Some corrections in Sects. 2.1, 2.2 were done. They don't reflect on
results and ideas. 31 pages, no figures. AMSTe
Hypermatrix factors for string and membrane junctions
The adjoint representations of the Lie algebras of the classical groups
SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric
products of two vector spaces, and hence are matrix representations. We
consider the analogous products of three vector spaces and study when they
appear as summands in Lie algebra decompositions. The Z3-grading of the
exceptional Lie algebras provide such summands and provides representations of
classical groups on hypermatrices. The main natural application is a formal
study of three-junctions of strings and membranes. Generalizations are also
considered.Comment: 25 pages, 4 figures, presentation improved, minor correction
A geometrical angle on Feynman integrals
A direct link between a one-loop N-point Feynman diagram and a geometrical
representation based on the N-dimensional simplex is established by relating
the Feynman parametric representations to the integrals over contents of
(N-1)-dimensional simplices in non-Euclidean geometry of constant curvature. In
particular, the four-point function in four dimensions is proportional to the
volume of a three-dimensional spherical (or hyperbolic) tetrahedron which can
be calculated by splitting into birectangular ones. It is also shown that the
known formula of reduction of the N-point function in (N-1) dimensions
corresponds to splitting the related N-dimensional simplex into N rectangular
ones.Comment: 47 pages, including 42 pages of the text (in plain Latex) and 5 pages
with the figures (in a separate Latex file, requires axodraw.sty) a note and
three references added, minor problem with notation fixe
Harmonic analysis on spherical homogeneous spaces with solvable stabilizer
For all spherical homogeneous spaces G/H, where G is a simply connected
semisimple algebraic group and H a connected solvable subgroup of G, we compute
the spectra of the representations of G on spaces of regular sections of
homogeneous line bundles over G/H.Comment: v2: 14 pages, minor correction
Local symmetry properties of pure 3-qubit states
Entanglement types of pure states of 3 qubits are classified by means of
their stabilisers in the group of local unitary operations. It is shown that
the stabiliser is generically discrete, and that a larger stabiliser indicates
a stationary value for some local invariant. We describe all the exceptional
states with enlarged stabilisers.Comment: 32 pages, 5 encapsulated PostScript files for 3 figures. Published
version, with minor correction
The orbit structure of Dynkin curves
Let G be a simple algebraic group over an algebraically closed field k;
assume that Char k is zero or good for G. Let \cB be the variety of Borel
subgroups of G and let e in Lie G be nilpotent. There is a natural action of
the centralizer C_G(e) of e in G on the Springer fibre \cB_e = {B' in \cB | e
in Lie B'} associated to e. In this paper we consider the case, where e lies in
the subregular nilpotent orbit; in this case \cB_e is a Dynkin curve. We give a
complete description of the C_G(e)-orbits in \cB_e. In particular, we classify
the irreducible components of \cB_e on which C_G(e) acts with finitely many
orbits. In an application we obtain a classification of all subregular orbital
varieties admitting a finite number of B-orbits for B a fixed Borel subgroup of
G.Comment: 12 pages, to appear in Math
A Unified Algebraic Approach to Classical Yang-Baxter Equation
In this paper, the different operator forms of classical Yang-Baxter equation
are given in the tensor expression through a unified algebraic method. It is
closely related to left-symmetric algebras which play an important role in many
fields in mathematics and mathematical physics. By studying the relations
between left-symmetric algebras and classical Yang-Baxter equation, we can
construct left-symmetric algebras from certain classical r-matrices and
conversely, there is a natural classical r-matrix constructed from a
left-symmetric algebra which corresponds to a parak\"ahler structure in
geometry. Moreover, the former in a special case gives an algebraic
interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a
classical r-matrix.Comment: To appear in Journal of Physics A: Mathematical and Theoretica
Geometric invariant theory approach to the determination of ground states of D-wave condensates in isotropic space
A complete and rigorous determination of the possible ground states for
D-wave pairing Bose condensates is presented, using a geometrical invariant
theory approach to the problem. The order parameter is argued to be a vector,
transforming according to a ten dimensional real representation of the group
{\bf O}{\bf U}. We determine the equalities
and inequalities defining the orbit space of this linear group and its symmetry
strata, which are in a one-to-one correspondence with the possible distinct
phases of the system. We find 15 allowed phases (besides the unbroken one),
with different symmetries, that we thoroughly determine. The group-subgroup
relations between bordering phases are pointed out. The perturbative sixth
degree corrections to the minimum of a fourth degree polynomial -invariant
free energy, calculated by Mermin, are also determined.Comment: 27 revtex pages, 2 figures, use of texdraw; minor changes in the
bibliography and in Table II
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