80,516 research outputs found
A quantum mechanical version of Price's theorem for Gaussian states
This paper is concerned with integro-differential identities which are known
in statistical signal processing as Price's theorem for expectations of
nonlinear functions of jointly Gaussian random variables. We revisit these
relations for classical variables by using the Frechet differentiation with
respect to covariance matrices, and then show that Price's theorem carries over
to a quantum mechanical setting. The quantum counterpart of the theorem is
established for Gaussian quantum states in the framework of the Weyl functional
calculus for quantum variables satisfying the Heisenberg canonical commutation
relations. The quantum mechanical version of Price's theorem relates the
Frechet derivative of the generalized moment of such variables with respect to
the real part of their quantum covariance matrix with other moments. As an
illustrative example, we consider these relations for quadratic-exponential
moments which are relevant to risk-sensitive quantum control.Comment: 11 pages, to appear in the Proceedings of the Australian Control
Conference, 17-18 November 2014, Canberra, Australi
Hessian and concavity of mutual information, differential entropy, and entropy power in linear vector Gaussian channels
Within the framework of linear vector Gaussian channels with arbitrary
signaling, closed-form expressions for the Jacobian of the minimum mean square
error and Fisher information matrices with respect to arbitrary parameters of
the system are calculated in this paper. Capitalizing on prior research where
the minimum mean square error and Fisher information matrices were linked to
information-theoretic quantities through differentiation, closed-form
expressions for the Hessian of the mutual information and the differential
entropy are derived. These expressions are then used to assess the concavity
properties of mutual information and differential entropy under different
channel conditions and also to derive a multivariate version of the entropy
power inequality due to Costa.Comment: 33 pages, 2 figures. A shorter version of this paper is to appear in
IEEE Transactions on Information Theor
A simple approach to counterterms in N=8 supergravity
We present a simple systematic method to study candidate counterterms in N=8
supergravity. Complicated details of the counterterm operators are avoided
because we work with the on-shell matrix elements they produce. All n-point
matrix elements of an independent SUSY invariant operator of the form D^{2k}
R^n +... must be local and satisfy SUSY Ward identities. These are strong
constraints, and we test directly whether or not matrix elements with these
properties can be constructed. If not, then the operator does not have a
supersymmetrization, and it is excluded as a potential counterterm. For n>4, we
find that R^n, D^2 R^n, D^4 R^n, and D^6 R^n are excluded as counterterms of
MHV amplitudes, while only R^n and D^2 R^n are excluded at the NMHV level. As a
consequence, for loop order L<7, there are no independent D^{2k}R^n
counterterms with n>4. If an operator is not ruled out, our method constructs
an explicit superamplitude for its matrix elements. This is done for the 7-loop
D^4 R^6 operator at the NMHV level and in other cases. We also initiate the
study of counterterms without leading pure-graviton matrix elements, which can
occur beyond the MHV level. The landscape of excluded/allowed candidate
counterterms is summarized in a colorful chart.Comment: 25 pages, 1 figure, published versio
Algebra of Observables for Identical Particles in One Dimension
The algebra of observables for identical particles on a line is formulated
starting from postulated basic commutation relations. A realization of this
algebra in the Calogero model was previously known. New realizations are
presented here in terms of differentiation operators and in terms of
SU(N)-invariant observables of the Hermitian matrix models. Some particular
structure properties of the algebra are briefly discussed.Comment: 13 pages, Latex, uses epsf, 1 eps figure include
Discrete transformation for matrix 3-waves problem in three dimensional space
Discrete transformation for 3- waves problem is constructed in explicit form.
Generalization of this system on the matrix case in three dimensional space
together with corresponding discrete transformation is presented also.Comment: LaTeX, 16 page
*-Structures on Braided Spaces
-structures on quantum and braided spaces of the type defined via an
R-matrix are studied. These include -Minkowski and -Euclidean spaces as
additive braided groups. The duality between the -braided groups of vectors
and covectors is proved and some first applications to braided geometry are
made.Comment: 20 page
Geometrical foundations of fractional supersymmetry
A deformed -calculus is developed on the basis of an algebraic structure
involving graded brackets. A number operator and left and right shift operators
are constructed for this algebra, and the whole structure is related to the
algebra of a -deformed boson. The limit of this algebra when is a -th
root of unity is also studied in detail. By means of a chain rule expansion,
the left and right derivatives are identified with the charge and covariant
derivative encountered in ordinary/fractional supersymmetry and this leads
to new results for these operators. A generalized Berezin integral and
fractional superspace measure arise as a natural part of our formalism. When
is a root of unity the algebra is found to have a non-trivial Hopf
structure, extending that associated with the anyonic line. One-dimensional
ordinary/fractional superspace is identified with the braided line when is
a root of unity, so that one-dimensional ordinary/fractional supersymmetry can
be viewed as invariance under translation along this line. In our construction
of fractional supersymmetry the -deformed bosons play a role exactly
analogous to that of the fermions in the familiar supersymmetric case.Comment: 42 pages, LaTeX. To appear in Int. J. Mod. Phys.
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