22,094 research outputs found
Semantics for a Quantum Programming Language by Operator Algebras
This paper presents a novel semantics for a quantum programming language by
operator algebras, which are known to give a formulation for quantum theory
that is alternative to the one by Hilbert spaces. We show that the opposite
category of the category of W*-algebras and normal completely positive
subunital maps is an elementary quantum flow chart category in the sense of
Selinger. As a consequence, it gives a denotational semantics for Selinger's
first-order functional quantum programming language QPL. The use of operator
algebras allows us to accommodate infinite structures and to handle classical
and quantum computations in a unified way.Comment: In Proceedings QPL 2014, arXiv:1412.810
Unsharp Values, Domains and Topoi
The so-called topos approach provides a radical reformulation of quantum
theory. Structurally, quantum theory in the topos formulation is very similar
to classical physics. There is a state object, analogous to the state space of
a classical system, and a quantity-value object, generalising the real numbers.
Physical quantities are maps from the state object to the quantity-value object
-- hence the `values' of physical quantities are not just real numbers in this
formalism. Rather, they are families of real intervals, interpreted as `unsharp
values'. We will motivate and explain these aspects of the topos approach and
show that the structure of the quantity-value object can be analysed using
tools from domain theory, a branch of order theory that originated in
theoretical computer science. Moreover, the base category of the topos
associated with a quantum system turns out to be a domain if the underlying von
Neumann algebra is a matrix algebra. For general algebras, the base category
still is a highly structured poset. This gives a connection between the topos
approach, noncommutative operator algebras and domain theory. In an outlook, we
present some early ideas on how domains may become useful in the search for new
models of (quantum) space and space-time.Comment: 32 pages, no figures; to appear in Proceedings of Quantum Field
Theory and Gravity, Regensburg (2010
Quantum Probability Theory
The mathematics of classical probability theory was subsumed into classical
measure theory by Kolmogorov in 1933. Quantum theory as nonclassical
probability theory was incorporated into the beginnings of noncommutative
measure theory by von Neumann in the early thirties, as well. To precisely this
end, von Neumann initiated the study of what are now called von Neumann
algebras and, with Murray, made a first classification of such algebras into
three types. The nonrelativistic quantum theory of systems with finitely many
degrees of freedom deals exclusively with type I algebras. However, for the
description of further quantum systems, the other types of von Neumann algebras
are indispensable. The paper reviews quantum probability theory in terms of
general von Neumann algebras, stressing the similarity of the conceptual
structure of classical and noncommutative probability theories and emphasizing
the correspondence between the classical and quantum concepts, though also
indicating the nonclassical nature of quantum probabilistic predictions. In
addition, differences between the probability theories in the type I, II and
III settings are explained. A brief description is given of quantum systems for
which probability theory based on type I algebras is known to be insufficient.
These illustrate the physical significance of the previously mentioned
differences.Comment: 28 pages, LaTeX, typos removed and some minor modifications for
clarity and accuracy made. This is the version to appear in Studies in the
History and Philosophy of Modern Physic
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