4,914 research outputs found
Geometry and symmetries of multi-particle systems
The quantum dynamical evolution of atomic and molecular aggregates, from
their compact to their fragmented states, is parametrized by a single
collective radial parameter. Treating all the remaining particle coordinates in
d dimensions democratically, as a set of angles orthogonal to this collective
radius or by equivalent variables, bypasses all independent-particle
approximations. The invariance of the total kinetic energy under arbitrary
d-dimensional transformations which preserve the radial parameter gives rise to
novel quantum numbers and ladder operators interconnecting its eigenstates at
each value of the radial parameter.
We develop the systematics and technology of this approach, introducing the
relevant mathematics tutorially, by analogy to the familiar theory of angular
momentum in three dimensions. The angular basis functions so obtained are
treated in a manifestly coordinate-free manner, thus serving as a flexible
generalized basis for carrying out detailed studies of wavefunction evolution
in multi-particle systems.Comment: 37 pages, 2 eps figure
General Bootstrap Equations in 4D CFTs
We provide a framework for generic 4D conformal bootstrap computations. It is
based on the unification of two independent approaches, the covariant
(embedding) formalism and the non-covariant (conformal frame) formalism. We
construct their main ingredients (tensor structures and differential operators)
and establish a precise connection between them. We supplement the discussion
by additional details like classification of tensor structures of n-point
functions, normalization of 2-point functions and seed conformal blocks,
Casimir differential operators and treatment of conserved operators and
permutation symmetries. Finally, we implement our framework in a Mathematica
package and make it freely available.Comment: 57 page
Rare-Event Sampling: Occupation-Based Performance Measures for Parallel Tempering and Infinite Swapping Monte Carlo Methods
In the present paper we identify a rigorous property of a number of
tempering-based Monte Carlo sampling methods, including parallel tempering as
well as partial and infinite swapping. Based on this property we develop a
variety of performance measures for such rare-event sampling methods that are
broadly applicable, informative, and straightforward to implement. We
illustrate the use of these performance measures with a series of applications
involving the equilibrium properties of simple Lennard-Jones clusters,
applications for which the performance levels of partial and infinite swapping
approaches are found to be higher than those of conventional parallel
tempering.Comment: 18 figure
Graph matching: relax or not?
We consider the problem of exact and inexact matching of weighted undirected
graphs, in which a bijective correspondence is sought to minimize a quadratic
weight disagreement. This computationally challenging problem is often relaxed
as a convex quadratic program, in which the space of permutations is replaced
by the space of doubly-stochastic matrices. However, the applicability of such
a relaxation is poorly understood. We define a broad class of friendly graphs
characterized by an easily verifiable spectral property. We prove that for
friendly graphs, the convex relaxation is guaranteed to find the exact
isomorphism or certify its inexistence. This result is further extended to
approximately isomorphic graphs, for which we develop an explicit bound on the
amount of weight disagreement under which the relaxation is guaranteed to find
the globally optimal approximate isomorphism. We also show that in many cases,
the graph matching problem can be further harmlessly relaxed to a convex
quadratic program with only n separable linear equality constraints, which is
substantially more efficient than the standard relaxation involving 2n equality
and n^2 inequality constraints. Finally, we show that our results are still
valid for unfriendly graphs if additional information in the form of seeds or
attributes is allowed, with the latter satisfying an easy to verify spectral
characteristic
Unambiguous discrimination among oracle operators
We address the problem of unambiguous discrimination among oracle operators.
The general theory of unambiguous discrimination among unitary operators is
extended with this application in mind. We prove that entanglement with an
ancilla cannot assist any discrimination strategy for commuting unitary
operators. We also obtain a simple, practical test for the unambiguous
distinguishability of an arbitrary set of unitary operators on a given system.
Using this result, we prove that the unambiguous distinguishability criterion
is the same for both standard and minimal oracle operators. We then show that,
except in certain trivial cases, unambiguous discrimination among all standard
oracle operators corresponding to integer functions with fixed domain and range
is impossible. However, we find that it is possible to unambiguously
discriminate among the Grover oracle operators corresponding to an arbitrarily
large unsorted database. The unambiguous distinguishability of standard oracle
operators corresponding to totally indistinguishable functions, which possess a
strong form of classical indistinguishability, is analysed. We prove that these
operators are not unambiguously distinguishable for any finite set of totally
indistinguishable functions on a Boolean domain and with arbitrary fixed range.
Sets of such functions on a larger domain can have unambiguously
distinguishable standard oracle operators and we provide a complete analysis of
the simplest case, that of four functions. We also examine the possibility of
unambiguous oracle operator discrimination with multiple parallel calls and
investigate an intriguing unitary superoperator transformation between standard
and entanglement-assisted minimal oracle operators.Comment: 35 pages. Final version. To appear in J. Phys. A: Math. & Theo
Design and analysis of bent functions using -subspaces
In this article, we provide the first systematic analysis of bent functions
on in the Maiorana-McFarland class
regarding the origin and cardinality of their -subspaces, i.e.,
vector subspaces on which the second-order derivatives of vanish. By
imposing restrictions on permutations of , we specify
the conditions, such that Maiorana-McFarland bent functions admit a unique -subspace of dimension . On the
other hand, we show that permutations with linear structures give rise to
Maiorana-McFarland bent functions that do not have this property. In this way,
we contribute to the classification of Maiorana-McFarland bent functions, since
the number of -subspaces is invariant under equivalence.
Additionally, we give several generic methods of specifying permutations
so that admits a unique -subspace. Most
notably, using the knowledge about -subspaces, we show that using
the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent
functions, one can in a generic manner generate bent functions on
outside the completed Maiorana-McFarland class
for any even . Remarkably, with our construction
methods it is possible to obtain inequivalent bent functions on
not stemming from two primary classes, the partial spread
class and . In this way, we contribute to a better
understanding of the origin of bent functions in eight variables, since only a
small fraction, of which size is about , stems from and
, whereas the total number of bent functions on
is approximately
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