326,729 research outputs found
Additive monotones for resource theories of parallel-combinable processes with discarding
A partitioned process theory, as defined by Coecke, Fritz, and Spekkens, is a
symmetric monoidal category together with an all-object-including symmetric
monoidal subcategory. We think of the morphisms of this category as processes,
and the morphisms of the subcategory as those processes that are freely
executable. Via a construction we refer to as parallel-combinable processes
with discarding, we obtain from this data a partially ordered monoid on the set
of processes, with f > g if one can use the free processes to construct g from
f. The structure of this partial order can then be probed using additive
monotones: order-preserving monoid homomorphisms with values in the real
numbers under addition. We first characterise these additive monotones in terms
of the corresponding partitioned process theory.
Given enough monotones, we might hope to be able to reconstruct the order on
the monoid. If so, we say that we have a complete family of monotones. In
general, however, when we require our monotones to be additive monotones, such
families do not exist or are hard to compute. We show the existence of complete
families of additive monotones for various partitioned process theories based
on the category of finite sets, in order to shed light on the way such families
can be constructed.Comment: In Proceedings QPL 2015, arXiv:1511.0118
Categorial L\'evy Processes
We generalize Franz' independence in tensor categories with inclusions from
two morphisms (which represent generalized random variables) to arbitrary
ordered families of morphisms. We will see that this only works consistently if
the unit object is an initial object, in which case the inclusions can be
defined starting from the tensor category alone. The obtained independence for
morphisms is called categorial independence. We define categorial L\'evy
processes on every tensor category with initial unit object and present a
construction generalizing the reconstruction of a L\'evy process from its
convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss
examples showing that many known independences from algebra as well as from
(noncommutative) probability are special cases of categorial independence.Comment: Changes in v2: Abstract and introduction extended. Background on
tensor functors moved to Section 2. Example section extended and reorganized.
References updated. Acknowledgements updated. (Some Enrivonment numbers have
changed!
Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models
In the categorical approach to the foundations of quantum theory, one begins
with a symmetric monoidal category, the objects of which represent physical
systems, and the morphisms of which represent physical processes. Usually, this
category is taken to be at least compact closed, and more often, dagger
compact, enforcing a certain self-duality, whereby preparation processes
(roughly, states) are inter-convertible with processes of registration
(roughly, measurement outcomes). This is in contrast to the more concrete
"operational" approach, in which the states and measurement outcomes associated
with a physical system are represented in terms of what we here call a "convex
operational model": a certain dual pair of ordered linear spaces -- generally,
{\em not} isomorphic to one another. On the other hand, state spaces for which
there is such an isomorphism, which we term {\em weakly self-dual}, play an
important role in reconstructions of various quantum-information theoretic
protocols, including teleportation and ensemble steering. In this paper, we
characterize compact closure of symmetric monoidal categories of convex
operational models in two ways: as a statement about the existence of
teleportation protocols, and as the principle that every process allowed by
that theory can be realized as an instance of a remote evaluation protocol ---
hence, as a form of classical probabilistic conditioning. In a large class of
cases, which includes both the classical and quantum cases, the relevant
compact closed categories are degenerate, in the weak sense that every object
is its own dual. We characterize the dagger-compactness of such a category
(with respect to the natural adjoint) in terms of the existence, for each
system, of a {\em symmetric} bipartite state, the associated conditioning map
of which is an isomorphism
Unitization during Category Learning
Five experiments explored the question of whether new perceptual units can be developed if they are diagnostic for a category learning task, and if so, what are the constraints on this unitization process? During category learning, participants were required to attend either a single component or a conjunction of five components in order to correctly categorize an object. In Experiments 1-4, some evidence for unitization was found in that the conjunctive task becomes much easier with practice, and this improvement was not found for the single component task, or for conjunctive tasks where the components cannot be unitized. Influences of component order (Experiment 1), component contiguity (Experiment 2), component proximity (Experiment 3), and number of components (Experiment 4) on practice effects were found. Using a Fourier Transformation method for deconvolving response times (Experiment 5), prolonged practice effects yielded responses that were faster than expected by analytic model that integrate evidence from independently perceived components
An alternative Gospel of structure: order, composition, processes
We survey some basic mathematical structures, which arguably are more
primitive than the structures taught at school. These structures are orders,
with or without composition, and (symmetric) monoidal categories. We list
several `real life' incarnations of each of these. This paper also serves as an
introduction to these structures and their current and potentially future uses
in linguistics, physics and knowledge representation.Comment: Introductory chapter to C. Heunen, M. Sadrzadeh, and E. Grefenstette.
Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse.
Oxford University Press, 201
Categorical Semantics for Functional Reactive Programming with Temporal Recursion and Corecursion
Functional reactive programming (FRP) makes it possible to express temporal
aspects of computations in a declarative way. Recently we developed two kinds
of categorical models of FRP: abstract process categories (APCs) and concrete
process categories (CPCs). Furthermore we showed that APCs generalize CPCs. In
this paper, we extend APCs with additional structure. This structure models
recursion and corecursion operators that are related to time. We show that the
resulting categorical models generalize those CPCs that impose an additional
constraint on time scales. This constraint boils down to ruling out
-supertasks, which are closely related to Zeno's paradox of Achilles
and the tortoise.Comment: In Proceedings MSFP 2014, arXiv:1406.153
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