Special Issue "Quantum Statistical Inference"

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A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (31 March 2015)

Special Issue Editor

Guest Editor
Dr. James D. Malley

Research Mathematical Statistician Building 12A, Room 2052 Mathematical and Statistical Computing Laboratory Division of Computational Bioscience Center for Information Technology National Institutes of Health Bethesda, MD 20892 USA
Phone: 301-496-9934

Special Issue Information

Dear Colleagues,

As quantum information theory advances, as new experimental quantum protocols are devised, and as the reality of practical quantum computing approaches, it is important to match these developments with appropriate statistical and data analysis methods. For this Special Issue the topic is understood to cover: advances in the probabilistic foundations of quantum mechanics; new developments for statistical methods and data analysis of quantum outcomes; quantum information perspectives on the analysis quantum outcomes and experiment; examination of classical statistical methods that have possible application to quantum outcomes and decisions, such as quantum based statistical learning machines, and Bayesian quantum state estimation. The emphasis in this Special Issue will be on experiment and applications of theory that advance our understanding of what quantum data can mean and how it might be used. More speculative revisions of the foundations of quantum mechanics are not obviously included in this discussion, unless grounded in current experiment or near term realizations of the proposed ideas and methods.

Dr. James D. Malley
Guest Editor

Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed Open Access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 300 CHF (Swiss Francs). English correction and/or formatting fees of 250 CHF (Swiss Francs) will be charged in certain cases for those articles accepted for publication that require extensive additional formatting and/or English corrections.

Published Papers (5 papers)

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Research

Open AccessArticle A Model for the Universe that Begins to Resemble a Quantum Computer
Axioms 2015, 4(1), 102-119; doi:10.3390/axioms4010102
Received: 10 December 2014 / Revised: 19 February 2015 / Accepted: 2 March 2015 / Published: 9 March 2015
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Abstract
This article presents a sequential growth model for the Universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a c-causet. A c-causet is defined to be a [...] Read more.
This article presents a sequential growth model for the Universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a c-causet. A c-causet is defined to be a causet that has a unique labeling. We characterize c-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different. We show that a c-causet has precisely two c-causet offspring. It follows that there are 2n c-causets of cardinality n + 1. This enables us to classify c-causets of cardinality n + 1 in terms of n-bits. We then quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the n-bits by n-qubits and defining transition amplitudes for the growth transitions. We mainly consider two types of processes, called stationary and completely stationary. We show that for stationary processes, the probability operators are tensor products of positive rank-one qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. We close with examples of precluded events. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)
Figures

Open AccessArticle Positive-Operator Valued Measure (POVM) Quantization
Axioms 2015, 4(1), 1-29; doi:10.3390/axioms4010001
Received: 3 September 2014 / Accepted: 18 December 2014 / Published: 25 December 2014
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Abstract
We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. [...] Read more.
We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple ormore elaborate examples illustrate the procedure: circle, two-sphere, plane and half-plane. Links with Positive-Operator Valued Measure (POVM) quantum measurement and quantum statistical inference are sketched. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)
Open AccessArticle Classical Probability and Quantum Outcomes
Axioms 2014, 3(2), 244-259; doi:10.3390/axioms3020244
Received: 2 April 2014 / Revised: 20 May 2014 / Accepted: 20 May 2014 / Published: 26 May 2014
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Abstract
There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this [...] Read more.
There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)
Open AccessCommunication Joint Distributions and Quantum Nonlocal Models
Axioms 2014, 3(2), 166-176; doi:10.3390/axioms3020166
Received: 2 December 2013 / Revised: 1 March 2014 / Accepted: 2 April 2014 / Published: 15 April 2014
Cited by 1 | PDF Full-text (164 KB) | HTML Full-text | XML Full-text
Abstract
A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state. A converse is demonstrated here: If a classical joint distribution for the pair agrees with standard quantum facts, then [...] Read more.
A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state. A converse is demonstrated here: If a classical joint distribution for the pair agrees with standard quantum facts, then the observables must commute. This has consequences for some historical and recent quantum nonlocal models: they are analytically disallowed without the need for experiment, as they imply that all local observables must commute among themselves. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)
Open AccessArticle Orthogonality and Dimensionality
Axioms 2013, 2(4), 477-489; doi:10.3390/axioms2040477
Received: 26 October 2013 / Revised: 28 November 2013 / Accepted: 10 December 2013 / Published: 13 December 2013
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Abstract
In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined [...] Read more.
In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formalism based on two basic ingredients, namely an orthogonality relation and matroids which are a very generic algebraic structure permitting to define a notion of dimension. Having obtained what we call orthomatroids, we then show that, in high enough dimension, the basic constituants of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbert lattices, so that their presence is a direct consequence of an orthogonality-based characterization of dimension. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)

Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Type of paper: Article
Title: The Universe as a Quantum Computer
Author: Stanley P. Gudder
Abstract: This article presents a sequential growth model for the universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a \(c\)-causet. A \(c\)-causet is defined to be a causet that is independent of its labeling. We characterize \(c\)-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different. We show that a \(c\)-causet has precisely two \(c\)-causet offspring. It follows that there are \(2^n\) \(c\)-causets of cardinality \(n+1\). This enables us to classify \(c\)-causets of cardinality \(n+1\) in terms of \(n\)-bits. We then quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the \(n\)-bits by \(n\)-qubits and defining transition amplitudes for the growth transitions. We mainly consider two types of processes called stationary and completely stationary. We show that for stationary processes, the probability operators are tensor products of positive rank-1 qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. We close with examples of precluded events.

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