7,378 research outputs found
Implications of quantum automata for contextuality
We construct zero-error quantum finite automata (QFAs) for promise problems
which cannot be solved by bounded-error probabilistic finite automata (PFAs).
Here is a summary of our results:
- There is a promise problem solvable by an exact two-way QFA in exponential
expected time, but not by any bounded-error sublogarithmic space probabilistic
Turing machine (PTM).
- There is a promise problem solvable by an exact two-way QFA in quadratic
expected time, but not by any bounded-error -space PTMs in
polynomial expected time. The same problem can be solvable by a one-way Las
Vegas (or exact two-way) QFA with quantum head in linear (expected) time.
- There is a promise problem solvable by a Las Vegas realtime QFA, but not by
any bounded-error realtime PFA. The same problem can be solvable by an exact
two-way QFA in linear expected time but not by any exact two-way PFA.
- There is a family of promise problems such that each promise problem can be
solvable by a two-state exact realtime QFAs, but, there is no such bound on the
number of states of realtime bounded-error PFAs solving the members this
family.
Our results imply that there exist zero-error quantum computational devices
with a \emph{single qubit} of memory that cannot be simulated by any finite
memory classical computational model. This provides a computational perspective
on results regarding ontological theories of quantum mechanics \cite{Hardy04},
\cite{Montina08}. As a consequence we find that classical automata based
simulation models \cite{Kleinmann11}, \cite{Blasiak13} are not sufficiently
powerful to simulate quantum contextuality. We conclude by highlighting the
interplay between results from automata models and their application to
developing a general framework for quantum contextuality.Comment: 22 page
Topological Foundations of Cognitive Science
A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers:
** Topological Foundations of Cognitive Science, Barry Smith
** The Bounds of Axiomatisation, Graham White
** Rethinking Boundaries, Wojciech Zelaniec
** Sheaf Mereology and Space Cognition, Jean Petitot
** A Mereotopological Definition of 'Point', Carola Eschenbach
** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel
** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda
** Defining a 'Doughnut' Made Difficult, N .M. Gotts
** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts
** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi
** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki
Zeno's Paradoxes. A Cardinal Problem 1. On Zenonian Plurality
In this paper the claim that Zeno's paradoxes have been solved is contested.
Although no one has ever touched Zeno without refuting him (Whitehead), it will
be our aim to show that, whatever it was that was refuted, it was certainly not
Zeno. The paper is organised in two parts. In the first part we will
demonstrate that upon direct analysis of the Greek sources, an underlying
structure common to both the Paradoxes of Plurality and the Paradoxes of Motion
can be exposed. This structure bears on a correct - Zenonian - interpretation
of the concept of division through and through. The key feature, generally
overlooked but essential to a correct understanding of all his arguments, is
that they do not presuppose time. Division takes place simultaneously. This
holds true for both PP and PM. In the second part a mathematical representation
will be set up that catches this common structure, hence the essence of all
Zeno's arguments, however without refuting them. Its central tenet is an
aequivalence proof for Zeno's procedure and Cantor's Continuum Hypothesis. Some
number theoretic and geometric implications will be shortly discussed.
Furthermore, it will be shown how the Received View on the motion-arguments can
easely be derived by the introduction of time as a (non-Zenonian) premiss, thus
causing their collapse into arguments which can be approached and refuted by
Aristotle's limit-like concept of the potentially infinite, which remained -
though in different disguises - at the core of the refutational strategies that
have been in use up to the present. Finally, an interesting link to Newtonian
mechanics via Cremona geometry can be established.Comment: 41 pages, 7 figure
Quantum Contextuality
A central result in the foundations of quantum mechanics is the
Kochen-Specker theorem. In short, it states that quantum mechanics is in
conflict with classical models in which the result of a measurement does not
depend on which other compatible measurements are jointly performed. Here,
compatible measurements are those that can be performed simultaneously or in
any order without disturbance. This conflict is generically called quantum
contextuality. In this article, we present an introduction to this subject and
its current status. We review several proofs of the Kochen-Specker theorem and
different notions of contextuality. We explain how to experimentally test some
of these notions and discuss connections between contextuality and nonlocality
or graph theory. Finally, we review some applications of contextuality in
quantum information processing.Comment: 63 pages, 20 figures. Updated version. Comments still welcome
Is the quantum state real? An extended review of -ontology theorems
Towards the end of 2011, Pusey, Barrett and Rudolph derived a theorem that
aimed to show that the quantum state must be ontic (a state of reality) in a
broad class of realist approaches to quantum theory. This result attracted a
lot of attention and controversy. The aim of this review article is to review
the background to the Pusey--Barrett--Rudolph Theorem, to provide a clear
presentation of the theorem itself, and to review related work that has
appeared since the publication of the Pusey--Barrett--Rudolph paper. In
particular, this review: Explains what it means for the quantum state to be
ontic or epistemic (a state of knowledge); Reviews arguments for and against an
ontic interpretation of the quantum state as they existed prior to the
Pusey--Barrett--Rudolph Theorem; Explains why proving the reality of the
quantum state is a very strong constraint on realist theories in that it would
imply many of the known no-go theorems, such as Bell's Theorem and the need for
an exponentially large ontic state space; Provides a comprehensive presentation
of the Pusey--Barrett--Rudolph Theorem itself, along with subsequent
improvements and criticisms of its assumptions; Reviews two other arguments for
the reality of the quantum state: the first due to Hardy and the second due to
Colbeck and Renner, and explains why their assumptions are less compelling than
those of the Pusey--Barrett--Rudolph Theorem; Reviews subsequent work aimed at
ruling out stronger notions of what it means for the quantum state to be
epistemic and points out open questions in this area. The overall aim is not
only to provide the background needed for the novice in this area to understand
the current status, but also to discuss often overlooked subtleties that should
be of interest to the experts.Comment: 88 pages, 15 figures, and a lot of sleepless nights. v2 is the
journal version. Reformatted in journal format, references updated, many typo
corrections and other minor updates. TeX source had to be modified slightly
to compile using the arXiv autocompiler, so I recommend downloading the
journal version from http://quanta.ws/ojs/index.php/quanta/article/view/2
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