4,468 research outputs found
Maximally epistemic interpretations of the quantum state and contextuality
We examine the relationship between quantum contextuality (in both the
standard Kochen-Specker sense and in the generalised sense proposed by
Spekkens) and models of quantum theory in which the quantum state is maximally
epistemic. We find that preparation noncontextual models must be maximally
epistemic, and these in turn must be Kochen-Specker noncontextual. This implies
that the Kochen-Specker theorem is sufficient to establish both the
impossibility of maximally epistemic models and the impossibility of
preparation noncontextual models. The implication from preparation
noncontextual to maximally epistemic then also yields a proof of Bell's theorem
from an EPR-like argument.Comment: v1: 4 pages, revTeX4.1, some overlap with arXiv:1207.7192. v2:
Changes in response to referees including revised proof of theorem 1, more
rigorous discussion of measure theoretic assumptions and extra introductory
materia
Complete Characterization of Functions Satisfying the Conditions of Arrow's Theorem
Arrow's theorem implies that a social choice function satisfying
Transitivity, the Pareto Principle (Unanimity) and Independence of Irrelevant
Alternatives (IIA) must be dictatorial. When non-strict preferences are
allowed, a dictatorial social choice function is defined as a function for
which there exists a single voter whose strict preferences are followed. This
definition allows for many different dictatorial functions. In particular, we
construct examples of dictatorial functions which do not satisfy Transitivity
and IIA. Thus Arrow's theorem, in the case of non-strict preferences, does not
provide a complete characterization of all social choice functions satisfying
Transitivity, the Pareto Principle, and IIA.
The main results of this article provide such a characterization for Arrow's
theorem, as well as for follow up results by Wilson. In particular, we
strengthen Arrow's and Wilson's result by giving an exact if and only if
condition for a function to satisfy Transitivity and IIA (and the Pareto
Principle). Additionally, we derive formulas for the number of functions
satisfying these conditions.Comment: 11 pages, 1 figur
Einstein-Podolsky-Rosen correlations and Bell correlations in the simplest scenario
Einstein-Podolsky-Rosen (EPR) steering is an intermediate type of quantum
nonlocality which sits between entanglement and Bell nonlocality. A set of
correlations is Bell nonlocal if it does not admit a local hidden variable
(LHV) model, while it is EPR nonlocal if it does not admit a local hidden
variable-local hidden state (LHV-LHS) model. It is interesting to know what
states can generate EPR-nonlocal correlations in the simplest nontrivial
scenario, that is, two projective measurements for each party sharing a
two-qubit state. Here we show that a two-qubit state can generate EPR-nonlocal
full correlations (excluding marginal statistics) in this scenario if and only
if it can generate Bell-nonlocal correlations. If full statistics (including
marginal statistics) is taken into account, surprisingly, the same scenario can
manifest the simplest one-way steering and the strongest hierarchy between
steering and Bell nonlocality. To illustrate these intriguing phenomena in
simple setups, several concrete examples are discussed in detail, which
facilitates experimental demonstration. In the course of study, we introduce
the concept of restricted LHS models and thereby derive a necessary and
sufficient semidefinite-programming criterion to determine the steerability of
any bipartite state under given measurements. Analytical criteria are further
derived in several scenarios of strong theoretical and experimental interest.Comment: New results added, 13 pages, 3 figures; published in Phys. Rev.
Positivity Problems for Low-Order Linear Recurrence Sequences
We consider two decision problems for linear recurrence sequences (LRS) over
the integers, namely the Positivity Problem (are all terms of a given LRS
positive?) and the Ultimate Positivity Problem} (are all but finitely many
terms of a given LRS positive?). We show decidability of both problems for LRS
of order 5 or less, with complexity in the Counting Hierarchy for Positivity,
and in polynomial time for Ultimate Positivity. Moreover, we show by way of
hardness that extending the decidability of either problem to LRS of order 6
would entail major breakthroughs in analytic number theory, more precisely in
the field of Diophantine approximation of transcendental numbers
Club guessing and the universal models
We survey the use of club guessing and other pcf constructs in the context of
showing that a given partially ordered class of objects does not have a
largest, or a universal element
Relational Hidden Variables and Non-Locality
We use a simple relational framework to develop the key notions and results
on hidden variables and non-locality. The extensive literature on these topics
in the foundations of quantum mechanics is couched in terms of probabilistic
models, and properties such as locality and no-signalling are formulated
probabilistically. We show that to a remarkable extent, the main structure of
the theory, through the major No-Go theorems and beyond, survives intact under
the replacement of probability distributions by mere relations.Comment: 42 pages in journal style. To appear in Studia Logic
A strictly stationary -mixing process satisfying the central limit theorem but not the weak invariance principle
In 1983, N. Herrndorf proved that for a -mixing sequence satisfying the
central limit theorem and , the weak
invariance principle takes place. The question whether for strictly stationary
sequences with finite second moments and a weaker type (, ,
) of mixing the central limit theorem implies the weak invariance
principle remained open.
We construct a strictly stationary -mixing sequence with finite
moments of any order and linear variance for which the central limit theorem
takes place but not the weak invariance principle.Comment: 12 page
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