Bart D'Hooghe

Bell-type inequalities in fuzzy probability calculus

Bell-type inequalities, used in mathematical physics as a criterion to check whether a physical situation allows description in terms of classical (Kolmogorovian) or quantum probability calculus are applied to various fuzzy probability models. It occurs that the standard set of Bell-type inequalities does not allow to distinguish Kolmogorovian probabilities from fuzzy probabilities based on the most frequently used Zadeh intersection or probabilistic intersection, but it allows to distinguish all these models from fuzzy probability models based on Giles (Łukasiewicz) intersection. It is proved that if we use fuzzy set intersections pointwisely generated by Franks fundamental triangular norms Ts(x,y), then the borderline between fuzzy probability models that can be distinguished from Kolmogorovian ones and these fuzzy probability models that cannot be distinguished is for .

A quantum cognition analysis of the Ellsberg paradox

The expected utility hypothesis is one of the foundations of classical approaches to economics and decision theory and Savages Sure-Thing Principle is a fundamental element of it. It has been put forward that real-life situations exist, illustrated by the Allais and Ellsberg paradoxes, in which the Sure-Thing Principle is violated, and where also the expected utility hypothesis does not hold. We have recently presented strong arguments for the presence of a double layer structure, a classical logical and a quantum conceptual, in human thought and that the quantum conceptual mode is responsible of the above violation. We consider in this paper the Ellsberg paradox, perform an experiment with real test subjects on the situation considered by Ellsberg, and use the collected data to elaborate a model for the conceptual landscape surrounding the decision situation of the paradox. We show that it is the overall conceptual landscape which gives rise to a violation of the Sure-Thing Principle and leads to the paradoxical situation discovered by Ellsberg.

On Some New Operations on Orthomodular Lattices

Kotas conditionals are used to define six pairs of disjunction- andconjunction-like operations on orthomodular lattices. Although five of them necessarily differfrom the lattice operations on elements that are not compatible, they coincidewith the lattice operations on all compatible elements of the lattice and theydefine on the underlying set a partial order relation that coincides with the originalone. Some of the new operations are noncommutative on noncompatible elements,but this does not exclude the possibility to endow them with a physicalinterpretation. The new operations are in general nonassociative, but for someof them a Foulis—Holland-type theorem concerning associativity instead ofdistributivity holds. The obtained results suggest that these new operations canserve as alternative algebraic models for the logical operations of disjunctionand conjunction.

A Geometrical Representation of Entanglement as Internal Constraint

We study a system of two entangled spin 1/2, were the spins are represented by a sphere model developed within the hidden measurement approach which is a generalization of the Bloch sphere representation, such that also the measurements are represented. We show how an arbitrary tensor product state can be described in a complete way by a specific internal constraint between the ray or density states of the two spin 1/2. We derive a geometrical view of entanglement as a “rotation” and “stretching” of the sphere representing the states of the second particle as measurements are performed on the first particle. In the case of the singlet state entanglement can be represented by a real physical constraint, namely by means of a rigid rod.

COMPATIBILITY AND SEPARABILITY FOR CLASSICAL AND QUANTUM ENTANGLEMENT

We study the concepts of compatibility and separability and their implications for quantum and classical systems. These concepts are illustrated on a macroscopic model for the singlet state of a quantum system of two entangled spin 1/2 with a parameter reflecting indeterminism in the measurement procedure. By varying this parameter we describe situations from quantum, intermediate to classical and study which tests are compatible or separated. We prove that for classical deterministic systems the concepts of separability and compatibility coincide, but for quantum systems and intermediate systems these concepts are generally different.

Classical Limit in Fuzzy Set Models of Spin-1/2 Quantum Logics

The classical limit of fuzzy set models ofspin-1/2 quantum logics is obtained in the course of a“defuzzyfication” procedure. The conditionsunder which the limiting structure is a Boolean algebra are studied.

Operator Structure of a Nonquantum and Nonclassical System

There exists a connection between the vectors of the Poincaré-sphere and the elements of the complex Hilbert space C2. This latter space is used to describe spin-1/2 measurements. We use this connection to study the intermediate cases of a more general spin-1/2 measurement model which has no representation in a Hilbert space. We construct the set of operators of this general model and investigate under which circumstances it is possible to define linear operators. Because no Hilbert space structure is possible for these intermediate cases, it can be expected that no linear operators are possible and it is shown that under very plausible assumptions this is indeed the case.

Simulating Quantum Computation on a Macroscopic Model

We present examples of macroscopic systems entailing a quantum mechanical structure. One of our examples has a structure which is isomorphic to the spin structure for a spin 1/2 and another system entails a structure isomorphic to the structure of two spin 1/2 in the entangled singlet state. We elaborate this system by showing that an arbitrary tensor product state representing two entangled qubits can be described in a complete way by a specific internal constraint between the ray or density states of the two qubits, which describes the behavior of the state of one of the spins if measurements are executed on the other spin. Since any n‐qubit unitary operation can be decomposed into 2‐qubit gates and unary operations, we argue that our representation of 2‐qubit entanglement contributes to a better understanding of the role of n‐qubit entanglement in quantum computation. We illustrate our approach on two 2‐qubit algorithms proposed by Deutsch, respectively Arvind et al.

QUANTUM COMPUTATION: TOWARDS THE CONSTRUCTION OF A ‘BETWEEN QUANTUM AND CLASSICAL’ COMPUTER

Using the ‘between quantum and classical’ models that have been constructed explicitly within the hidden measurement approach of quantum mechanics we investigate the possibility to construct a ‘between quantum and classical’ computer. In this view, the pure quantum computer and the classical Turing machine can be seen as two special cases of our general computer. We have shown in earlier research that the intermediate ‘between quantum and classical’ systems cannot be described within standard quantum theory. We argue that the general categoral approach of state property systems might provide a unified framework for the study of these ‘between quantum and classical’ models, and hence also for the study of classical and quantum computers as special cases.

INTERDISCIPLINARITY AND BRIDGING KNOWLEDGE

This volume is part of the ‘Worldviews, Science and Us’ series of proceedings and contains several contributions on the subject of bridging knowledge and its implications for our perspectives of the world. It represents the proceedings of the interdisciplinary stream of the international workshop Times of Entanglement, 21-22 September 2010 at the Minsheng Art Museum in Shanghai, People’s Republic of China in the context of the Shanghai World Expo 2010 and, related material from discussion panels organized by the Leo Apostel Center for Interdisciplinary studies within the framework of the ‘Research on the Construction of Integrating Worldviews’ research community set up by the Flanders Fund for Scientific Research. Further information about this research community and a full list of the associated international research centres can be found at http://www.vub.ac.be/CLEA/res/worldviews/ The first contribution to this volume, by Robrecht Vanderbeeken, is entitled ‘What about interdisciplinarity within philosophy?’. Vanderbeeken focuses on an ongoing split in contemporary Western philosophy between so-called continental philosophy and analytic philosophy. From a metaphilosophical stance, the common and prima facie response to this split is the encouragement of merging inclinations: a plea for unification in order to overcome the tension, to achieve a ‘jenseits’ of the distinction, a beyond, a post-condition. Vanderbeeken argues that there are good reasons to believe that unification coincides with a loss of authenticity and hence will blur the intellectual potential of both traditions. Vanderbeeken elaborates on these differences and takes them to be arguments in favour of a pluralist stance that resists the typical tendency of philosophers to bypass a theoretical impasse by means of a transgression to a new order.