7,290 research outputs found
Scott Continuity in Generalized Probabilistic Theories
Scott continuity is a concept from domain theory that had an unexpected
previous life in the theory of von Neumann algebras. Scott-continuous states
are known as normal states, and normal states are exactly the states coming
from density matrices. Given this, and the usefulness of Scott continuity in
domain theory, it is natural to ask whether this carries over to generalized
probabilistic theories. We show that the answer is no - there are
infinite-dimensional convex sets for which the set of Scott-continuous states
on the corresponding set of 2-valued POVMs does not recover the original convex
set, but is strictly larger. This shows the necessity of the use of topologies
for state-effect duality in the general case, rather than purely order
theoretic notions.Comment: In Proceedings QPL 2019, arXiv:2004.1475
Quantum state cloning using Deutschian closed timelike curves
We show that it is possible to clone quantum states to arbitrary accuracy in
the presence of a Deutschian closed timelike curve (D-CTC), with a fidelity
converging to one in the limit as the dimension of the CTC system becomes
large---thus resolving an open conjecture from [Brun et al., Physical Review
Letters 102, 210402 (2009)]. This result follows from a D-CTC-assisted scheme
for producing perfect clones of a quantum state prepared in a known eigenbasis,
and the fact that one can reconstruct an approximation of a quantum state from
empirical estimates of the probabilities of an informationally-complete
measurement. Our results imply more generally that every continuous, but
otherwise arbitrarily non-linear map from states to states can be implemented
to arbitrary accuracy with D-CTCs. Furthermore, our results show that Deutsch's
model for CTCs is in fact a classical model, in the sense that two arbitrary,
distinct density operators are perfectly distinguishable (in the limit of a
large CTC system); hence, in this model quantum mechanics becomes a classical
theory in which each density operator is a distinct point in a classical phase
space.Comment: 6 pages, 1 figure; v2: modifications to the interpretation of our
results based on the insightful comments of the referees; v3: minor change,
accepted for publication in Physical Review Letter
Nuclear and Trace Ideals in Tensored *-Categories
We generalize the notion of nuclear maps from functional analysis by defining
nuclear ideals in tensored *-categories. The motivation for this study came
from attempts to generalize the structure of the category of relations to
handle what might be called ``probabilistic relations''. The compact closed
structure associated with the category of relations does not generalize
directly, instead one obtains nuclear ideals. We introduce the notion of
nuclear ideal to analyze these classes of morphisms. In compact closed
categories, we see that all morphisms are nuclear, and in the category of
Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps.
We also introduce two new examples of tensored *-categories, in which
integration plays the role of composition. In the first, morphisms are a
special class of distributions, which we call tame distributions. We also
introduce a category of probabilistic relations which was the original
motivating example.
Finally, we extend the recent work of Joyal, Street and Verity on traced
monoidal categories to this setting by introducing the notion of a trace ideal.
For a given symmetric monoidal category, it is not generally the case that
arbitrary endomorphisms can be assigned a trace. However, we can find ideals in
the category on which a trace can be defined satisfying equations analogous to
those of Joyal, Street and Verity. We establish a close correspondence between
nuclear ideals and trace ideals in a tensored *-category, suggested by the
correspondence between Hilbert-Schmidt operators and trace operators on a
Hilbert space.Comment: 43 pages, Revised versio
Convergence and quantale-enriched categories
Generalising Nachbin's theory of "topology and order", in this paper we
continue the study of quantale-enriched categories equipped with a compact
Hausdorff topology. We compare these -categorical compact
Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that
the presence of a compact Hausdorff topology guarantees Cauchy completeness and
(suitably defined) codirected completeness of the underlying quantale enriched
category
ASSESSING FARMERS' ATTITUDES TOWARD RISK USING THE "CLOSING-IN" METHOD
The 1996 Farm Bill and low commodity prices have regenerated interest in the impact of risk and farmers' risk attitudes on production agriculture. Previous research has used expected utility theory (EUT) and direct elicitation of utility functions (DEU) for eliciting risk attitudes. To overcome the criticism of EUT and DEU, a recently developed technique called the "closing in" method is adapted for eliciting farmers' risk attitudes. This method is applied to Illinois farmers by using a computerized decision procedure, and is validated by comparing the results to the farmers' self-assessment of their risk attitudes and score to a risk attitudinal scale.Risk and Uncertainty,
Approximation in quantale-enriched categories
Our work is a fundamental study of the notion of approximation in
V-categories and in (U,V)-categories, for a quantale V and the ultrafilter
monad U. We introduce auxiliary, approximating and Scott-continuous
distributors, the way-below distributor, and continuity of V- and
(U,V)-categories. We fully characterize continuous V-categories (resp.
(U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in
the same ways as continuous domains are characterized among all dcpos. By
varying the choice of the quantale V and the notion of ideals, and by further
allowing the ultrafilter monad to act on the quantale, we obtain a flexible
theory of continuity that applies to partial orders and to metric and
topological spaces. We demonstrate on examples that our theory unifies some
major approaches to quantitative domain theory.Comment: 17 page
A categorical approach to the maximum theorem
Berge's maximum theorem gives conditions ensuring the continuity of an
optimised function as a parameter changes. In this paper we state and prove the
maximum theorem in terms of the theory of monoidal topology and the theory of
double categories.
This approach allows us to generalise (the main assertion of) the maximum
theorem, which is classically stated for topological spaces, to
pseudotopological spaces and pretopological spaces, as well as to closure
spaces, approach spaces and probabilistic approach spaces, amongst others. As a
part of this we prove a generalisation of the extreme value theorem.Comment: 45 pages. Minor changes in v2: this is the final preprint for
publication in JPA
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