14 research outputs found
On consistency of the quantum-like representation algorithm
In this paper we continue to study so called ``inverse Born's rule problem'':
to construct representation of probabilistic data of any origin by a complex
probability amplitude which matches Born's rule. The corresponding algorithm --
quantum-like representation algorithm (QLRA) was recently proposed by A.
Khrennikov [1]--[5]. Formally QLRA depends on the order of conditioning. For
two observables and - and conditional probabilities
produce two representations, say in Hilbert spaces and
In this paper we prove that under natural assumptions these two representations
are unitary equivalent. This result proves consistency QLRA
On the lattice structure of probability spaces in quantum mechanics
Let C be the set of all possible quantum states. We study the convex subsets
of C with attention focused on the lattice theoretical structure of these
convex subsets and, as a result, find a framework capable of unifying several
aspects of quantum mechanics, including entanglement and Jaynes' Max-Ent
principle. We also encounter links with entanglement witnesses, which leads to
a new separability criteria expressed in lattice language. We also provide an
extension of a separability criteria based on convex polytopes to the infinite
dimensional case and show that it reveals interesting facets concerning the
geometrical structure of the convex subsets. It is seen that the above
mentioned framework is also capable of generalization to any statistical theory
via the so-called convex operational models' approach. In particular, we show
how to extend the geometrical structure underlying entanglement to any
statistical model, an extension which may be useful for studying correlations
in different generalizations of quantum mechanics.Comment: arXiv admin note: substantial text overlap with arXiv:1008.416
Quantum Experimental Data in Psychology and Economics
We prove a theorem which shows that a collection of experimental data of
probabilistic weights related to decisions with respect to situations and their
disjunction cannot be modeled within a classical probabilistic weight structure
in case the experimental data contain the effect referred to as the
'disjunction effect' in psychology. We identify different experimental
situations in psychology, more specifically in concept theory and in decision
theory, and in economics (namely situations where Savage's Sure-Thing Principle
is violated) where the disjunction effect appears and we point out the common
nature of the effect. We analyze how our theorem constitutes a no-go theorem
for classical probabilistic weight structures for common experimental data when
the disjunction effect is affecting the values of these data. We put forward a
simple geometric criterion that reveals the non classicality of the considered
probabilistic weights and we illustrate our geometrical criterion by means of
experimentally measured membership weights of items with respect to pairs of
concepts and their disjunctions. The violation of the classical probabilistic
weight structure is very analogous to the violation of the well-known Bell
inequalities studied in quantum mechanics. The no-go theorem we prove in the
present article with respect to the collection of experimental data we consider
has a status analogous to the well known no-go theorems for hidden variable
theories in quantum mechanics with respect to experimental data obtained in
quantum laboratories. For this reason our analysis puts forward a strong
argument in favor of the validity of using a quantum formalism for modeling the
considered psychological experimental data as considered in this paper.Comment: 15 pages, 4 figure
Quantum Mechanics from Focusing and Symmetry
A foundation of quantum mechanics based on the concepts of focusing and
symmetry is proposed. Focusing is connected to c-variables - inaccessible
conceptually derived variables; several examples of such variables are given.
The focus is then on a maximal accessible parameter, a function of the common
c-variable. Symmetry is introduced via a group acting on the c-variable. From
this, the Hilbert space is constructed and state vectors and operators are
given a clear interpretation. The Born formula is proved from weak assumptions,
and from this the usual rules of quantum mechanics are derived. Several
paradoxes and other issues of quantum theory are discussed.Comment: 26 page
Quantum walks: a comprehensive review
Quantum walks, the quantum mechanical counterpart of classical random walks,
is an advanced tool for building quantum algorithms that has been recently
shown to constitute a universal model of quantum computation. Quantum walks is
now a solid field of research of quantum computation full of exciting open
problems for physicists, computer scientists, mathematicians and engineers.
In this paper we review theoretical advances on the foundations of both
discrete- and continuous-time quantum walks, together with the role that
randomness plays in quantum walks, the connections between the mathematical
models of coined discrete quantum walks and continuous quantum walks, the
quantumness of quantum walks, a summary of papers published on discrete quantum
walks and entanglement as well as a succinct review of experimental proposals
and realizations of discrete-time quantum walks. Furthermore, we have reviewed
several algorithms based on both discrete- and continuous-time quantum walks as
well as a most important result: the computational universality of both
continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing
Journa
Positive sets in finite linear function spaces
AbstractThis paper is mainly concerned with positive sets and positive functions ina finite linear function space. Our two main results characterize positive functions and minimal positive sets. We then show that certain morphisms preserve both the cone of positive functions and minimal positive sets. Finally we specialize these results to the case of measures on a hypergraph