5,137 research outputs found
Operators versus functions: from quantum dynamical semigroups to tomographic semigroups
Quantum mechanics can be formulated in terms of phase-space functions,
according to Wigner's approach. A generalization of this approach consists in
replacing the density operators of the standard formulation with suitable
functions, the so-called generalized Wigner functions or (group-covariant)
tomograms, obtained by means of group-theoretical methods. A typical problem
arising in this context is to express the evolution of a quantum system in
terms of tomograms. In the case of a (suitable) open quantum system, the
dynamics can be described by means of a quantum dynamical semigroup 'in
disguise', namely, by a semigroup of operators acting on tomograms rather than
on density operators. We focus on a special class of quantum dynamical
semigroups, the twirling semigroups, that have interesting applications, e.g.,
in quantum information science. The 'disguised counterparts' of the twirling
semigroups, i.e., the corresponding semigroups acting on tomograms, form a
class of semigroups of operators that we call tomographic semigroups. We show
that the twirling semigroups and the tomographic semigroups can be encompassed
in a unique theoretical framework, a class of semigroups of operators including
also the probability semigroups of classical probability theory, so achieving a
deeper insight into both the mathematical and the physical aspects of the
problem.Comment: 12 page
Star products: a group-theoretical point of view
Adopting a purely group-theoretical point of view, we consider the star
product of functions which is associated, in a natural way, with a square
integrable (in general, projective) representation of a locally compact group.
Next, we show that for this (implicitly defined) star product explicit formulae
can be provided. Two significant examples are studied in detail: the group of
translations on phase space and the one-dimensional affine group. The study of
the first example leads to the Groenewold-Moyal star product. In the second
example, the link with wavelet analysis is clarified.Comment: 42 pages; conclusions added; a few references adde
Discovering the manifold facets of a square integrable representation: from coherent states to open systems
Group representations play a central role in theoretical physics. In
particular, in quantum mechanics unitary --- or, in general, projective unitary
--- representations implement the action of an abstract symmetry group on
physical states and observables. More specifically, a major role is played by
the so-called square integrable representations. Indeed, the properties of
these representations are fundamental in the definition of certain families of
generalized coherent states, in the phase-space formulation of quantum
mechanics and the associated star product formalism, in the definition of an
interesting notion of function of quantum positive type, and in some recent
applications to the theory of open quantum systems and to quantum information.Comment: 13 page
Measurements of Electroweak Top Production at the LHC
The production of a single top quark at LHC may occur through charged-current
electroweak interactions, i.e. via three different processes: the s-channel
exchange of a virtual W boson, the t-channel exchange of a virtual W boson and
the associated production of a top quark and a W boson (tW). In this paper a
review of the measurements of the single-top production made both at ATLAS and
CMS will be shown.Comment: Proceedings of CKM 2012, the 7th International Workshop on the CKM
Unitarity Triangle, University of Cincinnati, USA, 28 September - 2 October
201
On the Complexity of ATL and ATL* Module Checking
Module checking has been introduced in late 1990s to verify open systems,
i.e., systems whose behavior depends on the continuous interaction with the
environment. Classically, module checking has been investigated with respect to
specifications given as CTL and CTL* formulas. Recently, it has been shown that
CTL (resp., CTL*) module checking offers a distinctly different perspective
from the better-known problem of ATL (resp., ATL*) model checking. In
particular, ATL (resp., ATL*) module checking strictly enhances the
expressiveness of both CTL (resp., CTL*) module checking and ATL (resp. ATL*)
model checking. In this paper, we provide asymptotically optimal bounds on the
computational cost of module checking against ATL and ATL*, whose upper bounds
are based on an automata-theoretic approach. We show that module-checking for
ATL is EXPTIME-complete, which is the same complexity of module checking
against CTL. On the other hand, ATL* module checking turns out to be
3EXPTIME-complete, hence exponentially harder than CTL* module checking.Comment: In Proceedings GandALF 2017, arXiv:1709.0176
Symmetry witnesses
A symmetry witness is a suitable subset of the space of selfadjoint trace
class operators that allows one to determine whether a linear map is a symmetry
transformation, in the sense of Wigner. More precisely, such a set is invariant
with respect to an injective densely defined linear operator in the Banach
space of selfadjoint trace class operators (if and) only if this operator is a
symmetry transformation. According to a linear version of Wigner's theorem, the
set of pure states, the rank-one projections, is a symmetry witness. We show
that an analogous result holds for the set of projections with a fixed rank
(with some mild constraint on this rank, in the finite-dimensional case). It
turns out that this result provides a complete classification of the set of
projections with a fixed rank that are symmetry witnesses. These particular
symmetry witnesses are projectable; i.e., reasoning in terms of quantum states,
the sets of uniform density operators of corresponding fixed rank are symmetry
witnesses too.Comment: 15 page
Reasoning about Knowledge and Strategies under Hierarchical Information
Two distinct semantics have been considered for knowledge in the context of
strategic reasoning, depending on whether players know each other's strategy or
not. The problem of distributed synthesis for epistemic temporal specifications
is known to be undecidable for the latter semantics, already on systems with
hierarchical information. However, for the other, uninformed semantics, the
problem is decidable on such systems. In this work we generalise this result by
introducing an epistemic extension of Strategy Logic with imperfect
information. The semantics of knowledge operators is uninformed, and captures
agents that can change observation power when they change strategies. We solve
the model-checking problem on a class of "hierarchical instances", which
provides a solution to a vast class of strategic problems with epistemic
temporal specifications on hierarchical systems, such as distributed synthesis
or rational synthesis
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