skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces

Abstract

The orthocomplemented modular lattice of subspaces L[H(d)], of a quantum system with d-dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H{sub 1},H{sub 2}), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H{sub 1}),P(H{sub 2}), to the subspaces H{sub 1}, H{sub 2}. As an application, it is shown that the proof of the inequalities of Clauser, Horne, Shimony, and Holt for a system of two spin 1/2 particles is valid for Kolmogorov probabilities, but it is not valid for Dempster-Shafer probabilities. The violation of these inequalities in experiments supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.

Authors:
 [1]
  1. Department of Computing, University of Bradford, Bradford BD7 1DP (United Kingdom)
Publication Date:
OSTI Identifier:
22306083
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; COMMUTATORS; FERMIONS; HILBERT SPACE; PROBABILITY; SPIN

Citation Formats

Vourdas, A. Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces. United States: N. p., 2014. Web. doi:10.1063/1.4891972.
Vourdas, A. Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces. United States. doi:10.1063/1.4891972.
Vourdas, A. Fri . "Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces". United States. doi:10.1063/1.4891972.
@article{osti_22306083,
title = {Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces},
author = {Vourdas, A.},
abstractNote = {The orthocomplemented modular lattice of subspaces L[H(d)], of a quantum system with d-dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H{sub 1},H{sub 2}), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H{sub 1}),P(H{sub 2}), to the subspaces H{sub 1}, H{sub 2}. As an application, it is shown that the proof of the inequalities of Clauser, Horne, Shimony, and Holt for a system of two spin 1/2 particles is valid for Kolmogorov probabilities, but it is not valid for Dempster-Shafer probabilities. The violation of these inequalities in experiments supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.},
doi = {10.1063/1.4891972},
journal = {Journal of Mathematical Physics},
number = 8,
volume = 55,
place = {United States},
year = {Fri Aug 15 00:00:00 EDT 2014},
month = {Fri Aug 15 00:00:00 EDT 2014}
}
  • A new method for multinomial inference is proposed by representing the cell probabilities as unordered segments on the unit interval and following Dempster-Shafer (DS) theory. The resulting DS posterior is then strengthened to improve symmetry and learning properties with the final posterior model being characterized by a Dirichlet distribution. In addition to computational simplicity, the new model has desirable invariance properties related to category permutations, refinements, and coarsenings. Furthemore, posterior inference on relative probabilities amongst certain cells depends only on data for the cells in question. Finally, the model is quite flexible with regard to parameterization and the range ofmore » testable assertions. Comparisons are made to existing methods and illustrated with two examples.« less
  • Identification and prioritization of risks to international security associated with emerging dual-use technologies presents numerous challenges. First, it demands prediction of the evolving states-of-the-art in various technological fields and, second, it requires a comprehension of the motivations and prospective selection criteria that illicit users might adopt in choosing among new technologies and their means of deployment. Nevertheless, the identification and prioritization of such threats is critical in establishing the appropriate focal points for proactive, nonproliferation policy-making. This paper addresses the question of how the threats associated with alternative means of deploying an emerging technology might be prioritized. The method revolvesmore » around systematic identification of the technological barriers to an illicit user in deploying a new technology. Evaluation of the resources necessary to overcome the barriers - such acquisition of the necessary intellectual capital and laboratory assets - then provides the basis for assessing the relative likelihoods or plausibilities of various deployment scenarios. Two optional bases are outlined for quantification of the model. One is a choice model that has found application in the analysis of consumer behavior, where the illicit user is modeled essentially as a consumer of new technology. The other employs a Dempster-Shafer framework for priority characterization. The paper describes application of the methodology to emerging life science technologies; in particular, to synthetic biology - the means of engineering biological systems. The prospect of a terrorist being able to synthesize natural pathogens or, perhaps worse still, to engineer pathogens not present in nature, creates an unprecedented threat to international security. Use of the proposed methodology to identify and prioritize threats associated with the engineering of pathogens is described.« less
  • Purpose: In cancer therapy, utilizing FDG-18 PET image-based features for accurate outcome prediction is challenging because of 1) limited discriminative information within a small number of PET image sets, and 2) fluctuant feature characteristics caused by the inferior spatial resolution and system noise of PET imaging. In this study, we proposed a new Dempster-Shafer theory (DST) based approach, evidential low-dimensional transformation with feature selection (ELT-FS), to accurately predict cancer therapy outcome with both PET imaging features and clinical characteristics. Methods: First, a specific loss function with sparse penalty was developed to learn an adaptive low-rank distance metric for representing themore » dissimilarity between different patients’ feature vectors. By minimizing this loss function, a linear low-dimensional transformation of input features was achieved. Also, imprecise features were excluded simultaneously by applying a l2,1-norm regularization of the learnt dissimilarity metric in the loss function. Finally, the learnt dissimilarity metric was applied in an evidential K-nearest-neighbor (EK- NN) classifier to predict treatment outcome. Results: Twenty-five patients with stage II–III non-small-cell lung cancer and thirty-six patients with esophageal squamous cell carcinomas treated with chemo-radiotherapy were collected. For the two groups of patients, 52 and 29 features, respectively, were utilized. The leave-one-out cross-validation (LOOCV) protocol was used for evaluation. Compared to three existing linear transformation methods (PCA, LDA, NCA), the proposed ELT-FS leads to higher prediction accuracy for the training and testing sets both for lung-cancer patients (100+/−0.0, 88.0+/−33.17) and for esophageal-cancer patients (97.46+/−1.64, 83.33+/−37.8). The ELT-FS also provides superior class separation in both test data sets. Conclusion: A novel DST- based approach has been proposed to predict cancer treatment outcome using PET image features and clinical characteristics. A specific loss function has been designed for robust accommodation of feature set incertitude and imprecision, facilitating adaptive learning of the dissimilarity metric for the EK-NN classifier.« less
  • This report summarizes methods to incorporate information (or lack of information) about inter-variable dependence into risk assessments that use Dempster-Shafer theory or probability bounds analysis to address epistemic and aleatory uncertainty. The report reviews techniques for simulating correlated variates for a given correlation measure and dependence model, computation of bounds on distribution functions under a specified dependence model, formulation of parametric and empirical dependence models, and bounding approaches that can be used when information about the intervariable dependence is incomplete. The report also reviews several of the most pervasive and dangerous myths among risk analysts about dependence in probabilistic models.
  • This report summarizes a variety of the most useful and commonly applied methods for obtaining Dempster-Shafer structures, and their mathematical kin probability boxes, from empirical information or theoretical knowledge. The report includes a review of the aggregation methods for handling agreement and conflict when multiple such objects are obtained from different sources.