A theory of concepts and their combinations II: A Hilbert space representation

Aerts, Diederik; Gabora, Liane

doi:10.1108/03684920510575807

Cited by 176 publications

(247 citation statements)

References 13 publications

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“…Applications of QP theory have been presented in decision making (Blutner et al, in press;Busemeyer, Wang, & Townsend, 2006;Bordley, 1998;Lambert-Mogiliansky, Zamir, & Zwirn, 2009;Pothos & Busemeyer, 2009;Wang & Busemeyer, in press;Yukalov & Sornette, 2010), conceptual combination (Aerts, 2009;Aerts & Gabora, 2005;Blutner, 2008;Bruza et al, under review), memory (Bruza, 2010;, and perception (Atmanspacher, Filk, & Romer, 2004). Psychological models based on quantum probability seem to work well (for overviews see Busemeyer & Bruza, 2009;Khrennikov, 2004;Pothos & Busemeyer, in press) and add to the increasing realization that the application of QP need not be restricted to physics.…”

Section: Introductionmentioning

confidence: 99%

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A quantum geometric model of similarity.

Pothos¹,

Busemeyer²,

Trueblood³

2013

Psychological Review

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This is the unspecified version of the paper.This version of the publication may differ from the final published version. Permanent repository link: AbstractNo other study has had as great an impact on the development of the similarity literature as that of Tversky (1977), which provided compelling demonstrations against all the fundamental assumptions of the popular, and extensively employed, geometric similarity models. Notably, similarity judgments were shown to violate symmetry and the triangle inequality, and also be subject to context effects, so that the same pair of items would be rated differently, depending on the presence of other items. Quantum theory provides a generalized geometric approach to similarity and can address several of Tversky's (1997) main findings. Similarity is modeled as quantum probability, so that asymmetries emerge as order effects, and the triangle equality violations and the diagnosticity effect can be related to the context-dependent properties of quantum probability. We so demonstrate the promise of the quantum approach for similarity and discuss the implications for representation theory in general.Keywords: similarity, metric axioms, symmetry, triangle inequality, diagnosticity, quantum probability 3 a quantum geometric model of similarity

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Section: Introductionmentioning

confidence: 99%

“…Quantum probability (QP) theory is a theory for how to assign probabilities to events (for more refined characterizations see e.g. Aerts & Gabora, 2005;Atmanspacher, Romer, & Wallach, 2006;Busemeyer & Bruza, 2012;Khrennikov, 2010). QP theory is a geometric theory of probability.…”

Section: Introductionmentioning

confidence: 99%

A quantum geometric model of similarity.

Pothos¹,

Busemeyer²,

Trueblood³

2013

Psychological Review

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“…Many of the relevant spaces considered in the field are of a very high dimension, and they do not need to satisfy the same set of physical conditions. Consider for example the very high dimensional cognitive spaces that are being modelled using QT [3,9]. We would not immediately expect such systems to display the same symmetry behaviour as a standard QT.…”

Section: Symmetry Groups For Quantum-like Theories?mentioning

confidence: 99%

“…This theorem shows that any mapping of a vector space onto itself that preserves the value of the inner product must be implemented by an operator U that is either unitary and linear or anti-unitary and anti-linear [25]. 3 Unitary operators are very widely used in QT, as they are the only ones that can describe continuous transformations such as translations and rotations (since every continuous transformation must have a square root [5]). However, anti-unitary transformations also play a part in the quantum formalism as they are used in the description of discrete time reversal symmetries.…”

Section: Transformations In Quantum Theorymentioning

confidence: 99%

Generalising Unitary Time Evolution

Kitto

Bruza

Sitbon

2009

Quantum Interaction

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Abstract. In this third Quantum Interaction (QI) meeting it is time to examine our failures. One of the weakest elements of QI as a field, arises in its continuing lack of models displaying proper evolutionary dynamics. This paper presents an overview of the modern generalised approach to the derivation of time evolution equations in physics, showing how the notion of symmetry is essential to the extraction of operators in quantum theory. The form that symmetry might take in non-physical models is explored, with a number of viable avenues identified. Quantum Interactions are not EvolvingAs a field Quantum Interaction (QI) has progressed well in recent years [10,8]. It is clear that something is to be gained from applying the quantum formalism to the description of systems not generally considered physical [1,4,14,16,23]. However, despite this initial promise, there are many elements of quantum theory that have yet to be properly applied within this framework. Perhaps most notably, it is clear that time evolution has yet to be properly implemented (i.e. derived) for any of these systems. This is a very significant weakness. Without an appreciation of how an entangled quantum-like system might come about it becomes rather difficult to justify the quantum collapse model that is very often leveraged in the quantum interaction community. This paper will explore the notion of time evolution in standard quantum theory (QT), sketching out the modern approach to extracting Hamiltonians and unitary operators. We shall then utilise this approach to suggest some interesting avenues that might be pursued in the future extraction of a fully-fledged quantum-like theory capable of evolving, entangling and then collapsing.There is no apriori reason to expect that the Schrödinger equation is the only form of time evolution equation available in a quantum-like theory. This paper will discuss the reasons lying behind this, and propose ways in which the QI community might work to establish a new time dynamics, or to prove that the application of Schrödinger dynamics is appropriate. Even if some justification can be found for the application of the Schrödinger equation beyond the description of physical systems, it is highly unlikely that the common techniques used in the extraction of a quantum description will work. This is because the standard approach to constructing a quantum theory generally involves finding a 2 Kirsty Kitto, Peter Bruza, and Laurianne Sitbon

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“…The rst one is related to disambiguating words [11]. A lattice of projectors associated to contexts is built, where atomic contexts (e.g.…”

Section: Qt and Irmentioning

confidence: 99%

Structured Information Retrieval and Quantum Theory

Piwowarski

Lalmas

2009

Quantum Interaction

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Abstract. Information Retrieval (IR) systems try to identify documents relevant to user queries, which are representations of user information needs. Interaction, context, and document structure are three important and active themes in IR research. We present how we propose to model the task of Structured IR (SIR) based on a QT inspired framework, with a focus on how to exploit user contextual information and user interaction in the search process.

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A theory of concepts and their combinations II: A Hilbert space representation

Cited by 176 publications

References 13 publications

A quantum geometric model of similarity.

A quantum geometric model of similarity.

Generalising Unitary Time Evolution

Structured Information Retrieval and Quantum Theory

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