154 research outputs found
Kochen-Specker Sets and Generalized Orthoarguesian Equations
Every set (finite or infinite) of quantum vectors (states) satisfies
generalized orthoarguesian equations (OA). We consider two 3-dim
Kochen-Specker (KS) sets of vectors and show how each of them should be
represented by means of a Hasse diagram---a lattice, an algebra of subspaces of
a Hilbert space--that contains rays and planes determined by the vectors so as
to satisfy OA. That also shows why they cannot be represented by a special
kind of Hasse diagram called a Greechie diagram, as has been erroneously done
in the literature. One of the KS sets (Peres') is an example of a lattice in
which 6OA pass and 7OA fails, and that closes an open question of whether the
7oa class of lattices properly contains the 6oa class. This result is important
because it provides additional evidence that our previously given proof of noa
=< (n+1)oa can be extended to proper inclusion noa < (n+1)oa and that nOA form
an infinite sequence of successively stronger equations.Comment: 16 pages and 5 figure
Corporate Memory
Corporate memory' is the body of information that an organization needs to keep for re-use. It is the active and historical information that an organization has that is worth sharing, managing and preserving to enable it to function effectively. This book is aimed at records managers and archivists, who are responsible for maintaining and managing information within an organization. It describes fully the most up-to-date methods and approaches to this essential function. In addition, it also discusses the adoption of an international standard for record management
New Class of 4-Dim Kochen-Specker Sets
We find a new highly symmetrical and very numerous class (millions of
non-isomorphic sets) of 4-dim Kochen-Specker (KS) vector sets. Due to the
nature of their geometrical symmetries, they cannot be obtained from previously
known ones. We generate the sets from a single set of 60 orthogonal spin
vectors and 75 of their tetrads (which we obtained from the 600-cell) by means
of our newly developed "stripping technique." We also consider "critical KS
subsets" and analyze their geometry. The algorithms and programs for the
generation of our KS sets are presented.Comment: 7 pages, 3 figures; to appear in J. Math. Phys. Vol.52, No. 2 (2011
Kochen-Specker Vectors
We give a constructive and exhaustive definition of Kochen-Specker (KS)
vectors in a Hilbert space of any dimension as well as of all the remaining
vectors of the space. KS vectors are elements of any set of orthonormal states,
i.e., vectors in n-dim Hilbert space, H^n, n>3 to which it is impossible to
assign 1s and 0s in such a way that no two mutually orthogonal vectors from the
set are both assigned 1 and that not all mutually orthogonal vectors are
assigned 0. Our constructive definition of such KS vectors is based on
algorithms that generate MMP diagrams corresponding to blocks of orthogonal
vectors in R^n, on algorithms that single out those diagrams on which algebraic
0-1 states cannot be defined, and on algorithms that solve nonlinear equations
describing the orthogonalities of the vectors by means of statistically
polynomially complex interval analysis and self-teaching programs. The
algorithms are limited neither by the number of dimensions nor by the number of
vectors. To demonstrate the power of the algorithms, all 4-dim KS vector
systems containing up to 24 vectors were generated and described, all 3-dim
vector systems containing up to 30 vectors were scanned, and several general
properties of KS vectors were found.Comment: 19 pages, 6 figures, title changed, introduction thoroughly
rewritten, n-dim rotation of KS vectors defined, original Kochen-Specker 192
(117) vector system translated into MMP diagram notation with a new graphical
representation, results on Tkadlec's dual diagrams added, several other new
results added, journal version: to be published in J. Phys. A, 38 (2005). Web
page: http://m3k.grad.hr/pavici
Hilbert Lattice Equations
There are five known classes of lattice equations that hold in every infinite
dimensional Hilbert space underlying quantum systems: generalised
orthoarguesian, Mayet's E_A, Godowski, Mayet-Godowski, and Mayet's E equations.
We obtain a result which opens a possibility that the first two classes
coincide. We devise new algorithms to generate Mayet-Godowski equations that
allow us to prove that the fourth class properly includes the third. An open
problem related to the last class is answered. Finally, we show some new
results on the Godowski lattices characterising the third class of equations.Comment: 24 pages, 3 figure
On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits
The Bell-Kochen-Specker theorem (BKS) theorem rules out realistic {\it
non-contextual} theories by resorting to impossible assignments of rays among a
selected set of maximal orthogonal bases. We investigate the geometrical
structure of small BKS-proofs involving real rays and
-dimensional bases of -qubits (). Specifically, we look at the
parity proof 18-9 with two qubits (A. Cabello, 1996), the parity proof 36-11
with three qubits (M. Kernaghan & A. Peres, 1995 \cite{Kernaghan1965}) and a
newly discovered non-parity proof 80-21 with four qubits (that improves work of
P. K Aravind's group in 2008). The rays in question arise as real eigenstates
shared by some maximal commuting sets (bases) of operators in the -qubit
Pauli group. One finds characteristic signatures of the distances between the
bases, which carry various symmetries in their graphs.Comment: version to appear in European Physical Journal Plu
Adding an Abstraction Barrier to ZF Set Theory
Much mathematical writing exists that is, explicitly or implicitly, based on
set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In
ZF, the domain of discourse contains only sets, and hence every mathematical
object must be a set. Consequently, in ZF, with the usual encoding of an
ordered pair , formulas like have truth values, and operations like have results that are sets. Such 'accidental theorems' do not match
how people think about the mathematics and also cause practical difficulties
when using set theory in machine-assisted theorem proving. In contrast, in a
number of proof assistants, mathematical objects and concepts can be built of
type-theoretic stuff so that many mathematical objects can be, in essence,
terms of an extended typed -calculus. However, dilemmas and
frustration arise when formalizing mathematics in type theory.
Motivated by problems of formalizing mathematics with (1) purely
set-theoretic and (2) type-theoretic approaches, we explore an option with much
of the flexibility of set theory and some of the useful features of type
theory. We present ZFP: a modification of ZF that has ordered pairs as
primitive, non-set objects. ZFP has a more natural and abstract axiomatic
definition of ordered pairs free of any notion of representation. This paper
presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of
the existence of a model for ZFP, which implies that ZFP is consistent if ZF
is. We discuss the approach used to add this abstraction barrier to ZF
Evaluation of elicitation methods to quantify Bayes linear models
The Bayes linear methodology allows decision makers to express their subjective beliefs and adjust these beliefs as observations are made. It is similar in spirit to probabilistic Bayesian approaches, but differs as it uses expectation as its primitive. While substantial work has been carried out in Bayes linear analysis, both in terms of theory development and application, there is little published material on the elicitation of structured expert judgement to quantify models. This paper investigates different methods that could be used by analysts when creating an elicitation process. The theoretical underpinnings of the elicitation methods developed are explored and an evaluation of their use is presented. This work was motivated by, and is a precursor to, an industrial application of Bayes linear modelling of the reliability of defence systems. An illustrative example demonstrates how the methods can be used in practice
Parity proofs of the Bell-Kochen-Specker theorem based on the 600-cell
The set of 60 real rays in four dimensions derived from the vertices of a
600-cell is shown to possess numerous subsets of rays and bases that provide
basis-critical parity proofs of the Bell-Kochen-Specker (BKS) theorem (a
basis-critical proof is one that fails if even a single basis is deleted from
it). The proofs vary considerably in size, with the smallest having 26 rays and
13 bases and the largest 60 rays and 41 bases. There are at least 90 basic
types of proofs, with each coming in a number of geometrically distinct
varieties. The replicas of all the proofs under the symmetries of the 600-cell
yield a total of almost a hundred million parity proofs of the BKS theorem. The
proofs are all very transparent and take no more than simple counting to
verify. A few of the proofs are exhibited, both in tabular form as well as in
the form of MMP hypergraphs that assist in their visualization. A survey of the
proofs is given, simple procedures for generating some of them are described
and their applications are discussed. It is shown that all four-dimensional
parity proofs of the BKS theorem can be turned into experimental disproofs of
noncontextuality.Comment: 19 pages, 11 tables, 3 figures. Email address of first author has
been corrected. Ref.[5] has been corrected, as has an error in Fig.3.
Formatting error in Sec.4 has been corrected and the placement of tables and
figures has been improved. A new paragraph has been added to Sec.4 and
another new paragraph to the end of the Appendi
Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions
It is pointed out that the 60 complex rays in four dimensions associated with
a system of two qubits yield over 10^9 critical parity proofs of the
Kochen-Specker theorem. The geometrical properties of the rays are described,
an overview of the parity proofs contained in them is given, and examples of
some of the proofs are exhibited.Comment: 17 pages, 13 tables, 3 figures. Several new references have been
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