55,114 research outputs found
Not throwing out the baby with the bathwater: Bell's condition of local causality mathematically 'sharp and clean'
The starting point of the present paper is Bell's notion of local causality
and his own sharpening of it so as to provide for mathematical formalisation.
Starting with Norsen's (2007, 2009) analysis of this formalisation, it is
subjected to a critique that reveals two crucial aspects that have so far not
been properly taken into account. These are (i) the correct understanding of
the notions of sufficiency, completeness and redundancy involved; and (ii) the
fact that the apparatus settings and measurement outcomes have very different
theoretical roles in the candidate theories under study. Both aspects are not
adequately incorporated in the standard formalisation, and we will therefore do
so. The upshot of our analysis is a more detailed, sharp and clean mathematical
expression of the condition of local causality. A preliminary analysis of the
repercussions of our proposal shows that it is able to locate exactly where and
how the notions of locality and causality are involved in formalising Bell's
condition of local causality.Comment: 14 pages. To be published in PSE volume "Explanation, Prediction, and
Confirmation", edited by Dieks, et a
Bell's theorem as a signature of nonlocality: a classical counterexample
For a system composed of two particles Bell's theorem asserts that averages
of physical quantities determined from local variables must conform to a family
of inequalities. In this work we show that a classical model containing a local
probabilistic interaction in the measurement process can lead to a violation of
the Bell inequalities. We first introduce two-particle phase-space
distributions in classical mechanics constructed to be the analogs of quantum
mechanical angular momentum eigenstates. These distributions are then employed
in four schemes characterized by different types of detectors measuring the
angular momenta. When the model includes an interaction between the detector
and the measured particle leading to ensemble dependencies, the relevant Bell
inequalities are violated if total angular momentum is required to be
conserved. The violation is explained by identifying assumptions made in the
derivation of Bell's theorem that are not fulfilled by the model. These
assumptions will be argued to be too restrictive to see in the violation of the
Bell inequalities a faithful signature of nonlocality.Comment: Extended manuscript. Significant change
The Identity Correspondence Problem and its Applications
In this paper we study several closely related fundamental problems for words
and matrices. First, we introduce the Identity Correspondence Problem (ICP):
whether a finite set of pairs of words (over a group alphabet) can generate an
identity pair by a sequence of concatenations. We prove that ICP is undecidable
by a reduction of Post's Correspondence Problem via several new encoding
techniques.
In the second part of the paper we use ICP to answer a long standing open
problem concerning matrix semigroups: "Is it decidable for a finitely generated
semigroup S of square integral matrices whether or not the identity matrix
belongs to S?". We show that the problem is undecidable starting from dimension
four even when the number of matrices in the generator is 48. From this fact,
we can immediately derive that the fundamental problem of whether a finite set
of matrices generates a group is also undecidable. We also answer several
question for matrices over different number fields. Apart from the application
to matrix problems, we believe that the Identity Correspondence Problem will
also be useful in identifying new areas of undecidable problems in abstract
algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the
published journal version of this article, see footnote 3 on page 1
Disentanglement and Decoherence by Open System Dynamics
The destruction of quantum interference, decoherence, and the destruction of
entanglement both appear to occur under the same circumstances. To address the
connection between these two phenomena, we consider the evolution of arbitrary
initial states of a two-particle system under open system dynamics described by
a class of master equations which produce decoherence of each particle. We show
that all initial states become separable after a finite time, and we produce
the explicit form of the separated state. The result extends and amplifies an
earlier result of Di\'osi. We illustrate the general result by considering the
case in which the initial state is an EPR state (in which both the positions
and momenta of a particle pair are perfectly correlated). This example clearly
illustrates how the spreading out in phase space produced by the environment
leads to certain disentanglement conditions becoming satisfied.Comment: 15 Page
Hardy's proof of nonlocality in the presence of noise
We extend the validity of Hardy's nonlocality without inequalities proof to
cover the case of special one-parameter classes of non-pure statistical
operators. These mixed states are obtained by mixing the Hardy states with a
completely chaotic noise or with a colored noise and they represent a realistic
description of imperfect preparation processes of (pure) Hardy states in
nonlocality experiments. Within such a framework we are able to exhibit a
precise range of values of the parameter measuring the noise affecting the
non-optimal preparation of an arbitrary Hardy state, for which it is still
possible to put into evidence genuine nonlocal effects. Equivalently, our work
exhibits particular classes of bipartite mixed states whose constituents do not
admit any local and deterministic hidden variable model reproducing the quantum
mechanical predictions.Comment: 9 pages, 2 figures, RevTex, revised versio
Spatial Economic Analysis in Data-Rich Environments
Controlling for spatial effects in micro-economic studies of consumer and producer behavior necessitates a range of analytical modifications ranging from modest changes in data collection and the definition of variables to dramatic changes in the modeling of consumer and producer decision-making. This paper discusses conceptual, empirical, and data issues involved in modeling the spatial aspects of economic behavior in data rich environments. Attention is given to established and emerging agricultural economic applications of spatial data and spatial econometric methods at the micro-scale. Recent applications of individual and household data are featured, including models of land-use change at the urban-rural interface, agricultural land values, and technological change and technology adoption.Research Methods/ Statistical Methods, C21, Q10, Q12, Q15, Q56,
Greenberger-Horne-Zeilinger argument of nonlocality without inequalities for mixed states
We generalize the Greenberger-Horne-Zeilinger nonlocality without
inequalities argument to cover the case of arbitrary mixed statistical
operators associated to three-qubits quantum systems. More precisely, we
determine the radius of a ball (in the trace distance topology) surrounding the
pure GHZ state and containing arbitrary mixed statistical operators which
cannot be described by any local and realistic hidden variable model and which
are, as a consequence, noncompletely separable. As a practical application, we
focus on certain one-parameter classes of mixed states which are commonly
considered in the experimental realization of the original GHZ argument and
which result from imperfect preparations of the pure GHZ state. In these cases
we determine for which values of the parameter controlling the noise a
nonlocality argument can still be exhibited, despite the mixedness of the
considered states. Moreover, the effect of the imperfect nature of measurement
processes is discussed.Comment: 8 pages, RevTex; added references, corrected typo
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