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Quantum stochastic walks on networks for decision-making.

Martínez-Martínez I, Sánchez-Burillo E - Sci Rep (2016)

Bottom Line: Evidence shows promise of a more general quantum theory providing a better explanation of the dynamics and structure of real decision-making processes than classical probability theory.Inspired by this, we show how the behavioral choice-probabilities can arise as the unique stationary distribution of quantum stochastic walkers on the classical network defined from Luce's response probabilities.We model the decision-maker as an open system in contact with her surrounding environment, and the time-length of the decision-making process reveals to be also a measure of the process' degree of interplay between the unitary and irreversible dynamics.

View Article: PubMed Central - PubMed

Affiliation: Düsseldorf Institute for Competition Economics (DICE), Heinrich Heine Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany.

ABSTRACT
Recent experiments report violations of the classical law of total probability and incompatibility of certain mental representations when humans process and react to information. Evidence shows promise of a more general quantum theory providing a better explanation of the dynamics and structure of real decision-making processes than classical probability theory. Inspired by this, we show how the behavioral choice-probabilities can arise as the unique stationary distribution of quantum stochastic walkers on the classical network defined from Luce's response probabilities. This work is relevant because (i) we provide a very general framework integrating the positive characteristics of both quantum and classical approaches previously in confrontation, and (ii) we define a cognitive network which can be used to bring other connectivist approaches to decision-making into the quantum stochastic realm. We model the decision-maker as an open system in contact with her surrounding environment, and the time-length of the decision-making process reveals to be also a measure of the process' degree of interplay between the unitary and irreversible dynamics. Implementing quantum coherence on classical networks may be a door to better integrate human-like reasoning biases in stochastic models for decision-making.

No MeSH data available.


Related in: MedlinePlus

Quantum stochastic walks.(a) Venn diagram showing the relationship between quantum walks (QWs–unitary evolution defined by the von Neumann equation) and classical random walks (CRWs–irreversible dynamics in a master equation) as two limiting cases within the family of quantum stochastic walks (QSWs) which includes more general probability distributions. We make use of the subset of QSWs interpolating between both cases through the parameter α ∈ [0, 1]. (b) Axiomatic construction of QSWs from an underlying graph. Each element of the connectivity matrix of the network corresponds to an allowed transition in the dynamical process. CRWs are defined from Axioms 1-2, QWs from Axioms 3-5, and Axiom 6 is only associated to those QSWs with no counterpart as just CRWs or QWs for their formalization. See Whitfield et al.36 for further reading.
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f1: Quantum stochastic walks.(a) Venn diagram showing the relationship between quantum walks (QWs–unitary evolution defined by the von Neumann equation) and classical random walks (CRWs–irreversible dynamics in a master equation) as two limiting cases within the family of quantum stochastic walks (QSWs) which includes more general probability distributions. We make use of the subset of QSWs interpolating between both cases through the parameter α ∈ [0, 1]. (b) Axiomatic construction of QSWs from an underlying graph. Each element of the connectivity matrix of the network corresponds to an allowed transition in the dynamical process. CRWs are defined from Axioms 1-2, QWs from Axioms 3-5, and Axiom 6 is only associated to those QSWs with no counterpart as just CRWs or QWs for their formalization. See Whitfield et al.36 for further reading.

Mentions: The second part in the master equation Eq. (2) contains the dissipative term responsible for the irreversibility in the decision-making process, weighted by the coefficient α such that the parameter α ∈ [0, 1] interpolates between the von Neumann evolution (α = 0) and the completely dissipative dynamics (α = 1). The section Methods covers the basics required to reach this formulation. See also Fig. 1 for an axiomatic construction of the quantum stochastic walks.


Quantum stochastic walks on networks for decision-making.

Martínez-Martínez I, Sánchez-Burillo E - Sci Rep (2016)

Quantum stochastic walks.(a) Venn diagram showing the relationship between quantum walks (QWs–unitary evolution defined by the von Neumann equation) and classical random walks (CRWs–irreversible dynamics in a master equation) as two limiting cases within the family of quantum stochastic walks (QSWs) which includes more general probability distributions. We make use of the subset of QSWs interpolating between both cases through the parameter α ∈ [0, 1]. (b) Axiomatic construction of QSWs from an underlying graph. Each element of the connectivity matrix of the network corresponds to an allowed transition in the dynamical process. CRWs are defined from Axioms 1-2, QWs from Axioms 3-5, and Axiom 6 is only associated to those QSWs with no counterpart as just CRWs or QWs for their formalization. See Whitfield et al.36 for further reading.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4815055&req=5

f1: Quantum stochastic walks.(a) Venn diagram showing the relationship between quantum walks (QWs–unitary evolution defined by the von Neumann equation) and classical random walks (CRWs–irreversible dynamics in a master equation) as two limiting cases within the family of quantum stochastic walks (QSWs) which includes more general probability distributions. We make use of the subset of QSWs interpolating between both cases through the parameter α ∈ [0, 1]. (b) Axiomatic construction of QSWs from an underlying graph. Each element of the connectivity matrix of the network corresponds to an allowed transition in the dynamical process. CRWs are defined from Axioms 1-2, QWs from Axioms 3-5, and Axiom 6 is only associated to those QSWs with no counterpart as just CRWs or QWs for their formalization. See Whitfield et al.36 for further reading.
Mentions: The second part in the master equation Eq. (2) contains the dissipative term responsible for the irreversibility in the decision-making process, weighted by the coefficient α such that the parameter α ∈ [0, 1] interpolates between the von Neumann evolution (α = 0) and the completely dissipative dynamics (α = 1). The section Methods covers the basics required to reach this formulation. See also Fig. 1 for an axiomatic construction of the quantum stochastic walks.

Bottom Line: Evidence shows promise of a more general quantum theory providing a better explanation of the dynamics and structure of real decision-making processes than classical probability theory.Inspired by this, we show how the behavioral choice-probabilities can arise as the unique stationary distribution of quantum stochastic walkers on the classical network defined from Luce's response probabilities.We model the decision-maker as an open system in contact with her surrounding environment, and the time-length of the decision-making process reveals to be also a measure of the process' degree of interplay between the unitary and irreversible dynamics.

View Article: PubMed Central - PubMed

Affiliation: Düsseldorf Institute for Competition Economics (DICE), Heinrich Heine Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany.

ABSTRACT
Recent experiments report violations of the classical law of total probability and incompatibility of certain mental representations when humans process and react to information. Evidence shows promise of a more general quantum theory providing a better explanation of the dynamics and structure of real decision-making processes than classical probability theory. Inspired by this, we show how the behavioral choice-probabilities can arise as the unique stationary distribution of quantum stochastic walkers on the classical network defined from Luce's response probabilities. This work is relevant because (i) we provide a very general framework integrating the positive characteristics of both quantum and classical approaches previously in confrontation, and (ii) we define a cognitive network which can be used to bring other connectivist approaches to decision-making into the quantum stochastic realm. We model the decision-maker as an open system in contact with her surrounding environment, and the time-length of the decision-making process reveals to be also a measure of the process' degree of interplay between the unitary and irreversible dynamics. Implementing quantum coherence on classical networks may be a door to better integrate human-like reasoning biases in stochastic models for decision-making.

No MeSH data available.


Related in: MedlinePlus