27 research outputs found
Characterization of quantum states in predicative logic
We develop a characterization of quantum states by means of first order
variables and random variables, within a predicative logic with equality, in
the framework of basic logic and its definitory equations. We introduce the
notion of random first order domain and find a characterization of pure states
in predicative logic and mixed states in propositional logic, due to a focusing
condition. We discuss the role of first order variables and the related
contextuality, in terms of sequents.Comment: 14 pages, Boston, IQSA10, to appea
A Topological Study of Contextuality and Modality in Quantum Mechanics
Kochen-Specker theorem rules out the non-contextual assignment of values to
physical magnitudes. Here we enrich the usual orthomodular structure of quantum
mechanical propositions with modal operators. This enlargement allows to refer
consistently to actual and possible properties of the system. By means of a
topological argument, more precisely in terms of the existence of sections of
sheaves, we give an extended version of Kochen-Specker theorem over this new
structure. This allows us to prove that contextuality remains a central feature
even in the enriched propositional system.Comment: 10 pages, no figures, submitted to I. J. Th. Phy
An Intrisic Topology for Orthomodular Lattices
We present a general way to define a topology on orthomodular lattices. We
show that in the case of a Hilbert lattice, this topology is equivalent to that
induced by the metrics of the corresponding Hilbert space. Moreover, we show
that in the case of a boolean algebra, the obtained topology is the discrete
one. Thus, our construction provides a general tool for studying orthomodular
lattices but also a way to distinguish classical and quantum logics.Comment: Under submission to the International Journal of Theoretical Physic
Partial Description of Quantum States
One of the most central and controversial element of quantum mechanics is the
use of non zero vectors of a Hilbert space (or, more generally, of one
dimension subspaces) for representing the state of a quantum system. In
particular, the question whether such a representation is complete has been
debated since almost the early days of quantum mechanics. In this article, we
develop an alternate way to formalize knowledge about the state of quantum
systems, based solely on experimentally accessible elements, namely on outcomes
of finite measurements. We introduce what we call partial description which,
given a feasible measurement, indicates some outcomes which are known to be
impossible (i.e. known to have a probability equal to 0 to occur) and hence
have to be discarded. Then, we introduce partial states (which are partial
descriptions providing as much information as possible) and compare this way to
describe quantum states to the orthodox one, using vector rays. Finally, we
show that partial states allow to describe quantum states in a strictly more
expressive way that the orthodox description does
Generalised quantum weakest preconditions
Generalisation of the quantum weakest precondition result of D'Hondt and
Panangaden is presented. In particular the most general notion of quantum
predicate as positive operator valued measure (POVM) is introduced. The
previously known quantum weakest precondition result has been extended to cover
the case of POVM playing the role of a quantum predicate. Additionally, our
result is valid in infinite dimension case and also holds for a quantum
programs defined as a positive but not necessary completely positive
transformations of a quantum states.Comment: 7 pages, no figures, added references, changed conten
On the lattice structure of probability spaces in quantum mechanics
Let C be the set of all possible quantum states. We study the convex subsets
of C with attention focused on the lattice theoretical structure of these
convex subsets and, as a result, find a framework capable of unifying several
aspects of quantum mechanics, including entanglement and Jaynes' Max-Ent
principle. We also encounter links with entanglement witnesses, which leads to
a new separability criteria expressed in lattice language. We also provide an
extension of a separability criteria based on convex polytopes to the infinite
dimensional case and show that it reveals interesting facets concerning the
geometrical structure of the convex subsets. It is seen that the above
mentioned framework is also capable of generalization to any statistical theory
via the so-called convex operational models' approach. In particular, we show
how to extend the geometrical structure underlying entanglement to any
statistical model, an extension which may be useful for studying correlations
in different generalizations of quantum mechanics.Comment: arXiv admin note: substantial text overlap with arXiv:1008.416
`What is a Thing?': Topos Theory in the Foundations of Physics
The goal of this paper is to summarise the first steps in developing a
fundamentally new way of constructing theories of physics. The motivation comes
from a desire to address certain deep issues that arise when contemplating
quantum theories of space and time. In doing so we provide a new answer to
Heidegger's timeless question ``What is a thing?''.
Our basic contention is that constructing a theory of physics is equivalent
to finding a representation in a topos of a certain formal language that is
attached to the system. Classical physics uses the topos of sets. Other
theories involve a different topos. For the types of theory discussed in this
paper, a key goal is to represent any physical quantity with an arrow
\breve{A}_\phi:\Si_\phi\map\R_\phi where \Si_\phi and are two
special objects (the `state-object' and `quantity-value object') in the
appropriate topos, .
We discuss two different types of language that can be attached to a system,
. The first, \PL{S}, is a propositional language; the second, \L{S}, is
a higher-order, typed language. Both languages provide deductive systems with
an intuitionistic logic. With the aid of \PL{S} we expand and develop some of
the earlier work (By CJI and collaborators.) on topos theory and quantum
physics. A key step is a process we term `daseinisation' by which a projection
operator is mapped to a sub-object of the spectral presheaf \Sig--the topos
quantum analogue of a classical state space. The topos concerned is \SetH{}:
the category of contravariant set-valued functors on the category (partially
ordered set) \V{} of commutative sub-algebras of the algebra of bounded
operators on the quantum Hilbert space \Hi.Comment: To appear in ``New Structures in Physics'' ed R. Coeck
A many-valued approach to quantum computational logics
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Quantum computational logics are special examples of quantum logic where formulas are supposed to denote pieces of quantum information (qubit-systems or mixtures of qubit-systems), while logical connectives are interpreted as reversible quantum logical gates. Hence, any formula of the quantum computational language represents a synthetic logical description of a quantum circuit. We investigate a many-valued approach to quantum information, where the basic notion of qubit has been replaced by the more general notion of qudit. The qudit-semantics allows us to represent as reversible gates some basic logical operations of Ćukasiewicz many-valued logics. In the ïŹnal part of the article we discuss some problems that concern possible implementations of gates by means of optical device